# Math

extern class java.lang.MathAvailable in javaThe class {@code Math} contains methods for performing basic numeric operations such as the elementary exponential, logarithm, square root, and trigonometric functions.

<p>Unlike some of the numeric methods of class {@code StrictMath}, all implementations of the equivalent functions of class {@code Math} are not defined to return the bit-for-bit same results. This relaxation permits better-performing implementations where strict reproducibility is not required.

<p>By default many of the {@code Math} methods simply call the equivalent method in {@code StrictMath} for their implementation. Code generators are encouraged to use platform-specific native libraries or microprocessor instructions, where available, to provide higher-performance implementations of {@code Math} methods. Such higher-performance implementations still must conform to the specification for {@code Math}.

<p>The quality of implementation specifications concern two properties, accuracy of the returned result and monotonicity of the method. Accuracy of the floating-point {@code Math} methods is measured in terms of <i>ulps</i>, units in the last place. For a given floating-point format, an ulp of a specific real number value is the distance between the two floating-point values bracketing that numerical value. When discussing the accuracy of a method as a whole rather than at a specific argument, the number of ulps cited is for the worst-case error at any argument. If a method always has an error less than 0.5 ulps, the method always returns the floating-point number nearest the exact result; such a method is <i>correctly rounded</i>. A correctly rounded method is generally the best a floating-point approximation can be; however, it is impractical for many floating-point methods to be correctly rounded. Instead, for the {@code Math} class, a larger error bound of 1 or 2 ulps is allowed for certain methods. Informally, with a 1 ulp error bound, when the exact result is a representable number, the exact result should be returned as the computed result; otherwise, either of the two floating-point values which bracket the exact result may be returned. For exact results large in magnitude, one of the endpoints of the bracket may be infinite. Besides accuracy at individual arguments, maintaining proper relations between the method at different arguments is also important. Therefore, most methods with more than 0.5 ulp errors are required to be <i>semi-monotonic</i>: whenever the mathematical function is non-decreasing, so is the floating-point approximation, likewise, whenever the mathematical function is non-increasing, so is the floating-point approximation. Not all approximations that have 1 ulp accuracy will automatically meet the monotonicity requirements.

author Joseph D. Darcy @since JDK1.0 static var E(default,null) : FloatThe {@code double} value that is closer than any other to <i>e</i>, the base of the natural logarithms. static var PI(default,null) : FloatThe {@code double} value that is closer than any other to <i>pi</i>, the ratio of the circumference of a circle to its diameter. static function IEEEremainder( f1 : Float, f2 : Float ) : FloatComputes the remainder operation on two arguments as prescribed by the IEEE 754 standard. The remainder value is mathematically equal to

param f2 the divisor. @return the remainder when {@code f1} is divided by {@code f2}. static function abs( a : Int ) : IntReturns the absolute value of an {@code int} value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned.

<p>Note that if the argument is equal to the value of {@link Integer#MIN_VALUE}, the most negative representable {@code int} value, the result is that same value, which is negative.

return the absolute value of the argument. static function acos( a : Float ) : FloatReturns the arc cosine of a value; the returned angle is in the range 0.0 through <i>pi</i>. Special case: <ul><li>If the argument is NaN or its absolute value is greater than 1, then the result is NaN.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the arc cosine of the argument. static function asin( a : Float ) : FloatReturns the arc sine of a value; the returned angle is in the range -<i>pi</i>/2 through <i>pi</i>/2. Special cases: <ul><li>If the argument is NaN or its absolute value is greater than 1, then the result is NaN. <li>If the argument is zero, then the result is a zero with the same sign as the argument.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the arc sine of the argument. static function atan( a : Float ) : FloatReturns the arc tangent of a value; the returned angle is in the range -<i>pi</i>/2 through <i>pi</i>/2. Special cases: <ul><li>If the argument is NaN, then the result is NaN. <li>If the argument is zero, then the result is a zero with the same sign as the argument.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the arc tangent of the argument. static function atan2( y : Float, x : Float ) : FloatReturns the angle <i>theta</i> from the conversion of rectangular coordinates ({@code x}, {@code y}) to polar coordinates (r, <i>theta</i>). This method computes the phase <i>theta</i> by computing an arc tangent of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special cases: <ul><li>If either argument is NaN, then the result is NaN. <li>If the first argument is positive zero and the second argument is positive, or the first argument is positive and finite and the second argument is positive infinity, then the result is positive zero. <li>If the first argument is negative zero and the second argument is positive, or the first argument is negative and finite and the second argument is positive infinity, then the result is negative zero. <li>If the first argument is positive zero and the second argument is negative, or the first argument is positive and finite and the second argument is negative infinity, then the result is the {@code double} value closest to <i>pi</i>. <li>If the first argument is negative zero and the second argument is negative, or the first argument is negative and finite and the second argument is negative infinity, then the result is the {@code double} value closest to -<i>pi</i>. <li>If the first argument is positive and the second argument is positive zero or negative zero, or the first argument is positive infinity and the second argument is finite, then the result is the {@code double} value closest to <i>pi</i>/2. <li>If the first argument is negative and the second argument is positive zero or negative zero, or the first argument is negative infinity and the second argument is finite, then the result is the {@code double} value closest to -<i>pi</i>/2. <li>If both arguments are positive infinity, then the result is the {@code double} value closest to <i>pi</i>/4. <li>If the first argument is positive infinity and the second argument is negative infinity, then the result is the {@code double} value closest to 3*<i>pi</i>/4. <li>If the first argument is negative infinity and the second argument is positive infinity, then the result is the {@code double} value closest to -<i>pi</i>/4. <li>If both arguments are negative infinity, then the result is the {@code double} value closest to -3*<i>pi</i>/4.</ul>

<p>The computed result must be within 2 ulps of the exact result. Results must be semi-monotonic.

param x the abscissa coordinate @return the <i>theta</i> component of the point (<i>r</i>, <i>theta</i>) in polar coordinates that corresponds to the point (<i>x</i>, <i>y</i>) in Cartesian coordinates. static function cbrt( a : Float ) : FloatReturns the cube root of a {@code double} value. For positive finite {@code x}, {@code cbrt(-x) == -cbrt(x)}; that is, the cube root of a negative value is the negative of the cube root of that value's magnitude.

Special cases:

<ul>

<li>If the argument is NaN, then the result is NaN.

<li>If the argument is infinite, then the result is an infinity with the same sign as the argument.

<li>If the argument is zero, then the result is a zero with the same sign as the argument.

</ul>

<p>The computed result must be within 1 ulp of the exact result.

return the cube root of {@code a}. @since 1.5 static function ceil( a : Float ) : FloatReturns the smallest (closest to negative infinity) {@code double} value that is greater than or equal to the argument and is equal to a mathematical integer. Special cases: <ul><li>If the argument value is already equal to a mathematical integer, then the result is the same as the argument. <li>If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument. <li>If the argument value is less than zero but greater than -1.0, then the result is negative zero.</ul> Note that the value of {@code Math.ceil(x)} is exactly the value of {@code -Math.floor(-x)}.

return the smallest (closest to negative infinity) floating-point value that is greater than or equal to the argument and is equal to a mathematical integer. static function copySign( magnitude : Float, sign : Float ) : FloatReturns the first floating-point argument with the sign of the second floating-point argument. Note that unlike the {@link StrictMath#copySign(double, double) StrictMath.copySign} method, this method does not require NaN {@code sign} arguments to be treated as positive values; implementations are permitted to treat some NaN arguments as positive and other NaN arguments as negative to allow greater performance.

param sign the parameter providing the sign of the result @return a value with the magnitude of {@code magnitude} and the sign of {@code sign}. @since 1.6 static function cos( a : Float ) : FloatReturns the trigonometric cosine of an angle. Special cases: <ul><li>If the argument is NaN or an infinity, then the result is NaN.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

@param a an angle, in radians. @return the cosine of the argument. static function cosh( x : Float ) : FloatReturns the hyperbolic cosine of a {@code double} value. The hyperbolic cosine of <i>x</i> is defined to be (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2 where <i>e</i> is {@linkplain Math#E Euler's number}.

<p>Special cases: <ul>

<li>If the argument is NaN, then the result is NaN.

<li>If the argument is infinite, then the result is positive infinity.

<li>If the argument is zero, then the result is {@code 1.0}.

</ul>

<p>The computed result must be within 2.5 ulps of the exact result.

return The hyperbolic cosine of {@code x}. @since 1.5 static function exp( a : Float ) : FloatReturns Euler's number <i>e</i> raised to the power of a {@code double} value. Special cases: <ul><li>If the argument is NaN, the result is NaN. <li>If the argument is positive infinity, then the result is positive infinity. <li>If the argument is negative infinity, then the result is positive zero.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

@param a the exponent to raise <i>e</i> to. @return the value <i>e</i><sup>{@code a}</sup>, where <i>e</i> is the base of the natural logarithms. static function expm1( x : Float ) : FloatReturns <i>e</i><sup>x</sup> -1. Note that for values of <i>x</i> near 0, the exact sum of {@code expm1(x)} + 1 is much closer to the true result of <i>e</i><sup>x</sup> than {@code exp(x)}.

<p>Special cases: <ul> <li>If the argument is NaN, the result is NaN.

<li>If the argument is positive infinity, then the result is positive infinity.

<li>If the argument is negative infinity, then the result is -1.0.

<li>If the argument is zero, then the result is a zero with the same sign as the argument.

</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic. The result of {@code expm1} for any finite input must be greater than or equal to {@code -1.0}. Note that once the exact result of <i>e</i><sup>{@code x}</sup> - 1 is within 1/2 ulp of the limit value -1, {@code -1.0} should be returned.

@param x the exponent to raise <i>e</i> to in the computation of <i>e</i><sup>{@code x}</sup> -1. @return the value <i>e</i><sup>{@code x}</sup> - 1. @since 1.5 static function floor( a : Float ) : FloatReturns the largest (closest to positive infinity) {@code double} value that is less than or equal to the argument and is equal to a mathematical integer. Special cases: <ul><li>If the argument value is already equal to a mathematical integer, then the result is the same as the argument. <li>If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument.</ul>

return the largest (closest to positive infinity) floating-point value that less than or equal to the argument and is equal to a mathematical integer. static function getExponent( f : Single ) : IntReturns the unbiased exponent used in the representation of a {@code float}. Special cases:

<ul> <li>If the argument is NaN or infinite, then the result is {@link Float#MAX_EXPONENT} + 1. <li>If the argument is zero or subnormal, then the result is {@link Float#MIN_EXPONENT} -1. </ul> @param f a {@code float} value since 1.6 static function hypot( x : Float, y : Float ) : FloatReturns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) without intermediate overflow or underflow.

<p>Special cases: <ul>

<li> If either argument is infinite, then the result is positive infinity.

<li> If either argument is NaN and neither argument is infinite, then the result is NaN.

</ul>

<p>The computed result must be within 1 ulp of the exact result. If one parameter is held constant, the results must be semi-monotonic in the other parameter.

param y a value @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) without intermediate overflow or underflow @since 1.5 static function log( a : Float ) : FloatReturns the natural logarithm (base <i>e</i>) of a {@code double} value. Special cases: <ul><li>If the argument is NaN or less than zero, then the result is NaN. <li>If the argument is positive infinity, then the result is positive infinity. <li>If the argument is positive zero or negative zero, then the result is negative infinity.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the value ln {@code a}, the natural logarithm of {@code a}. static function log10( a : Float ) : FloatReturns the base 10 logarithm of a {@code double} value. Special cases:

<ul><li>If the argument is NaN or less than zero, then the result is NaN. <li>If the argument is positive infinity, then the result is positive infinity. <li>If the argument is positive zero or negative zero, then the result is negative infinity. <li> If the argument is equal to 10<sup><i>n</i></sup> for integer <i>n</i>, then the result is <i>n</i>. </ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the base 10 logarithm of {@code a}. @since 1.5 static function log1p( x : Float ) : FloatReturns the natural logarithm of the sum of the argument and 1. Note that for small values {@code x}, the result of {@code log1p(x)} is much closer to the true result of ln(1 + {@code x}) than the floating-point evaluation of {@code log(1.0+x)}.

<p>Special cases:

<ul>

<li>If the argument is NaN or less than -1, then the result is NaN.

<li>If the argument is positive infinity, then the result is positive infinity.

<li>If the argument is negative one, then the result is negative infinity.

<li>If the argument is zero, then the result is a zero with the same sign as the argument.

</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the value ln({@code x} + 1), the natural log of {@code x} + 1 @since 1.5 static function max( a : Int, b : Int ) : IntReturns the greater of two {@code int} values. That is, the result is the argument closer to the value of {@link Integer#MAX_VALUE}. If the arguments have the same value, the result is that same value.

param b another argument. @return the larger of {@code a} and {@code b}. static function min( a : Int, b : Int ) : IntReturns the smaller of two {@code int} values. That is, the result the argument closer to the value of {@link Integer#MIN_VALUE}. If the arguments have the same value, the result is that same value.

param b another argument. @return the smaller of {@code a} and {@code b}. static function nextAfter( start : Float, direction : Float ) : FloatReturns the floating-point number adjacent to the first argument in the direction of the second argument. If both arguments compare as equal the second argument is returned.

<p> Special cases: <ul> <li> If either argument is a NaN, then NaN is returned.

<li> If both arguments are signed zeros, {@code direction} is returned unchanged (as implied by the requirement of returning the second argument if the arguments compare as equal).

<li> If {@code start} is ±{@link Double#MIN_VALUE} and {@code direction} has a value such that the result should have a smaller magnitude, then a zero with the same sign as {@code start} is returned.

<li> If {@code start} is infinite and {@code direction} has a value such that the result should have a smaller magnitude, {@link Double#MAX_VALUE} with the same sign as {@code start} is returned.

<li> If {@code start} is equal to ± {@link Double#MAX_VALUE} and {@code direction} has a value such that the result should have a larger magnitude, an infinity with same sign as {@code start} is returned. </ul>

param direction value indicating which of {@code start}'s neighbors or {@code start} should be returned @return The floating-point number adjacent to {@code start} in the direction of {@code direction}. @since 1.6 static function nextUp( d : Float ) : FloatReturns the floating-point value adjacent to {@code d} in the direction of positive infinity. This method is semantically equivalent to {@code nextAfter(d, Double.POSITIVE_INFINITY)}; however, a {@code nextUp} implementation may run faster than its equivalent {@code nextAfter} call.

<p>Special Cases: <ul> <li> If the argument is NaN, the result is NaN.

<li> If the argument is positive infinity, the result is positive infinity.

<li> If the argument is zero, the result is {@link Double#MIN_VALUE}

</ul>

return The adjacent floating-point value closer to positive infinity. @since 1.6 static function pow( a : Float, b : Float ) : FloatReturns the value of the first argument raised to the power of the second argument. Special cases:

<ul><li>If the second argument is positive or negative zero, then the result is 1.0. <li>If the second argument is 1.0, then the result is the same as the first argument. <li>If the second argument is NaN, then the result is NaN. <li>If the first argument is NaN and the second argument is nonzero, then the result is NaN.

<li>If <ul> <li>the absolute value of the first argument is greater than 1 and the second argument is positive infinity, or <li>the absolute value of the first argument is less than 1 and the second argument is negative infinity, </ul> then the result is positive infinity.

<li>If <ul> <li>the absolute value of the first argument is greater than 1 and the second argument is negative infinity, or <li>the absolute value of the first argument is less than 1 and the second argument is positive infinity, </ul> then the result is positive zero.

<li>If the absolute value of the first argument equals 1 and the second argument is infinite, then the result is NaN.

<li>If <ul> <li>the first argument is positive zero and the second argument is greater than zero, or <li>the first argument is positive infinity and the second argument is less than zero, </ul> then the result is positive zero.

<li>If <ul> <li>the first argument is positive zero and the second argument is less than zero, or <li>the first argument is positive infinity and the second argument is greater than zero, </ul> then the result is positive infinity.

<li>If <ul> <li>the first argument is negative zero and the second argument is greater than zero but not a finite odd integer, or <li>the first argument is negative infinity and the second argument is less than zero but not a finite odd integer, </ul> then the result is positive zero.

<li>If <ul> <li>the first argument is negative zero and the second argument is a positive finite odd integer, or <li>the first argument is negative infinity and the second argument is a negative finite odd integer, </ul> then the result is negative zero.

<li>If <ul> <li>the first argument is negative zero and the second argument is less than zero but not a finite odd integer, or <li>the first argument is negative infinity and the second argument is greater than zero but not a finite odd integer, </ul> then the result is positive infinity.

<li>If <ul> <li>the first argument is negative zero and the second argument is a negative finite odd integer, or <li>the first argument is negative infinity and the second argument is a positive finite odd integer, </ul> then the result is negative infinity.

<li>If the first argument is finite and less than zero <ul> <li> if the second argument is a finite even integer, the result is equal to the result of raising the absolute value of the first argument to the power of the second argument

<li>if the second argument is a finite odd integer, the result is equal to the negative of the result of raising the absolute value of the first argument to the power of the second argument

<li>if the second argument is finite and not an integer, then the result is NaN. </ul>

<li>If both arguments are integers, then the result is exactly equal to the mathematical result of raising the first argument to the power of the second argument if that result can in fact be represented exactly as a {@code double} value.</ul>

<p>(In the foregoing descriptions, a floating-point value is considered to be an integer if and only if it is finite and a fixed point of the method {@link #ceil ceil} or, equivalently, a fixed point of the method {@link #floor floor}. A value is a fixed point of a one-argument method if and only if the result of applying the method to the value is equal to the value.)

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

param b the exponent. @return the value {@code a}<sup>{@code b}</sup>. static function random() : FloatReturns a {@code double} value with a positive sign, greater than or equal to {@code 0.0} and less than {@code 1.0}. Returned values are chosen pseudorandomly with (approximately) uniform distribution from that range.

<p>When this method is first called, it creates a single new pseudorandom-number generator, exactly as if by the expression

<blockquote>{@code new java.util.Random()}</blockquote>

This new pseudorandom-number generator is used thereafter for all calls to this method and is used nowhere else.

<p>This method is properly synchronized to allow correct use by more than one thread. However, if many threads need to generate pseudorandom numbers at a great rate, it may reduce contention for each thread to have its own pseudorandom-number generator.

@return a pseudorandom {@code double} greater than or equal to {@code 0.0} and less than {@code 1.0}. @see Random#nextDouble() static function rint( a : Float ) : FloatReturns the {@code double} value that is closest in value to the argument and is equal to a mathematical integer. If two {@code double} values that are mathematical integers are equally close, the result is the integer value that is even. Special cases: <ul><li>If the argument value is already equal to a mathematical integer, then the result is the same as the argument. <li>If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument.</ul>

@param a a {@code double} value. @return the closest floating-point value to {@code a} that is equal to a mathematical integer. static function round( a : Single ) : IntReturns the closest {@code int} to the argument, with ties rounding up.

<p> Special cases: <ul><li>If the argument is NaN, the result is 0. <li>If the argument is negative infinity or any value less than or equal to the value of {@code Integer.MIN_VALUE}, the result is equal to the value of {@code Integer.MIN_VALUE}. <li>If the argument is positive infinity or any value greater than or equal to the value of {@code Integer.MAX_VALUE}, the result is equal to the value of {@code Integer.MAX_VALUE}.</ul>

return the value of the argument rounded to the nearest {@code int} value. @see java.lang.Integer#MAX_VALUE @see java.lang.Integer#MIN_VALUE static function scalb( d : Float, scaleFactor : Int ) : FloatReturn {@code d} × 2<sup>{@code scaleFactor}</sup> rounded as if performed by a single correctly rounded floating-point multiply to a member of the double value set. See the Java Language Specification for a discussion of floating-point value sets. If the exponent of the result is between {@link Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the answer is calculated exactly. If the exponent of the result would be larger than {@code Double.MAX_EXPONENT}, an infinity is returned. Note that if the result is subnormal, precision may be lost; that is, when {@code scalb(x, n)} is subnormal, {@code scalb(scalb(x, n), -n)} may not equal <i>x</i>. When the result is non-NaN, the result has the same sign as {@code d}.

<p>Special cases: <ul> <li> If the first argument is NaN, NaN is returned. <li> If the first argument is infinite, then an infinity of the same sign is returned. <li> If the first argument is zero, then a zero of the same sign is returned. </ul>

param scaleFactor power of 2 used to scale {@code d} @return {@code d} × 2<sup>{@code scaleFactor}</sup> @since 1.6 static function signum( d : Float ) : FloatReturns the signum function of the argument; zero if the argument is zero, 1.0 if the argument is greater than zero, -1.0 if the argument is less than zero.

<p>Special Cases: <ul> <li> If the argument is NaN, then the result is NaN. <li> If the argument is positive zero or negative zero, then the result is the same as the argument. </ul>

return the signum function of the argument since 1.5 static function sin( a : Float ) : FloatReturns the trigonometric sine of an angle. Special cases: <ul><li>If the argument is NaN or an infinity, then the result is NaN. <li>If the argument is zero, then the result is a zero with the same sign as the argument.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

@param a an angle, in radians. @return the sine of the argument. static function sinh( x : Float ) : FloatReturns the hyperbolic sine of a {@code double} value. The hyperbolic sine of <i>x</i> is defined to be (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2 where <i>e</i> is {@linkplain Math#E Euler's number}.

<p>Special cases: <ul>

<li>If the argument is NaN, then the result is NaN.

<li>If the argument is infinite, then the result is an infinity with the same sign as the argument.

<li>If the argument is zero, then the result is a zero with the same sign as the argument.

</ul>

<p>The computed result must be within 2.5 ulps of the exact result.

return The hyperbolic sine of {@code x}. @since 1.5 static function sqrt( a : Float ) : FloatReturns the correctly rounded positive square root of a {@code double} value. Special cases: <ul><li>If the argument is NaN or less than zero, then the result is NaN. <li>If the argument is positive infinity, then the result is positive infinity. <li>If the argument is positive zero or negative zero, then the result is the same as the argument.</ul> Otherwise, the result is the {@code double} value closest to the true mathematical square root of the argument value.

return the positive square root of {@code a}. If the argument is NaN or less than zero, the result is NaN. static function tan( a : Float ) : FloatReturns the trigonometric tangent of an angle. Special cases: <ul><li>If the argument is NaN or an infinity, then the result is NaN. <li>If the argument is zero, then the result is a zero with the same sign as the argument.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

@param a an angle, in radians. @return the tangent of the argument. static function tanh( x : Float ) : FloatReturns the hyperbolic tangent of a {@code double} value. The hyperbolic tangent of <i>x</i> is defined to be (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>), in other words, {@linkplain Math#sinh sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note that the absolute value of the exact tanh is always less than 1.

<p>Special cases: <ul>

<li>If the argument is NaN, then the result is NaN.

<li>If the argument is zero, then the result is a zero with the same sign as the argument.

<li>If the argument is positive infinity, then the result is {@code +1.0}.

<li>If the argument is negative infinity, then the result is {@code -1.0}.

</ul>

<p>The computed result must be within 2.5 ulps of the exact result. The result of {@code tanh} for any finite input must have an absolute value less than or equal to 1. Note that once the exact result of tanh is within 1/2 of an ulp of the limit value of ±1, correctly signed ±{@code 1.0} should be returned.

return The hyperbolic tangent of {@code x}. @since 1.5 static function toDegrees( angrad : Float ) : FloatConverts an angle measured in radians to an approximately equivalent angle measured in degrees. The conversion from radians to degrees is generally inexact; users should <i>not</i> expect {@code cos(toRadians(90.0))} to exactly equal {@code 0.0}.

@param angrad an angle, in radians @return the measurement of the angle {@code angrad} in degrees. @since 1.2 static function toRadians( angdeg : Float ) : FloatConverts an angle measured in degrees to an approximately equivalent angle measured in radians. The conversion from degrees to radians is generally inexact.

@param angdeg an angle, in degrees @return the measurement of the angle {@code angdeg} in radians. @since 1.2 static function ulp( d : Float ) : FloatReturns the size of an ulp of the argument. An ulp of a {@code double} value is the positive distance between this floating-point value and the {@code double} value next larger in magnitude. Note that for non-NaN <i>x</i>,

<p>Special Cases: <ul> <li> If the argument is NaN, then the result is NaN. <li> If the argument is positive or negative infinity, then the result is positive infinity. <li> If the argument is positive or negative zero, then the result is {@code Double.MIN_VALUE}. <li> If the argument is ±{@code Double.MAX_VALUE}, then the result is equal to 2<sup>971</sup>. </ul>

return the size of an ulp of the argument since 1.5

<p>Unlike some of the numeric methods of class {@code StrictMath}, all implementations of the equivalent functions of class {@code Math} are not defined to return the bit-for-bit same results. This relaxation permits better-performing implementations where strict reproducibility is not required.

<p>By default many of the {@code Math} methods simply call the equivalent method in {@code StrictMath} for their implementation. Code generators are encouraged to use platform-specific native libraries or microprocessor instructions, where available, to provide higher-performance implementations of {@code Math} methods. Such higher-performance implementations still must conform to the specification for {@code Math}.

<p>The quality of implementation specifications concern two properties, accuracy of the returned result and monotonicity of the method. Accuracy of the floating-point {@code Math} methods is measured in terms of <i>ulps</i>, units in the last place. For a given floating-point format, an ulp of a specific real number value is the distance between the two floating-point values bracketing that numerical value. When discussing the accuracy of a method as a whole rather than at a specific argument, the number of ulps cited is for the worst-case error at any argument. If a method always has an error less than 0.5 ulps, the method always returns the floating-point number nearest the exact result; such a method is <i>correctly rounded</i>. A correctly rounded method is generally the best a floating-point approximation can be; however, it is impractical for many floating-point methods to be correctly rounded. Instead, for the {@code Math} class, a larger error bound of 1 or 2 ulps is allowed for certain methods. Informally, with a 1 ulp error bound, when the exact result is a representable number, the exact result should be returned as the computed result; otherwise, either of the two floating-point values which bracket the exact result may be returned. For exact results large in magnitude, one of the endpoints of the bracket may be infinite. Besides accuracy at individual arguments, maintaining proper relations between the method at different arguments is also important. Therefore, most methods with more than 0.5 ulp errors are required to be <i>semi-monotonic</i>: whenever the mathematical function is non-decreasing, so is the floating-point approximation, likewise, whenever the mathematical function is non-increasing, so is the floating-point approximation. Not all approximations that have 1 ulp accuracy will automatically meet the monotonicity requirements.

author Joseph D. Darcy @since JDK1.0 static var E(default,null) : FloatThe {@code double} value that is closer than any other to <i>e</i>, the base of the natural logarithms. static var PI(default,null) : FloatThe {@code double} value that is closer than any other to <i>pi</i>, the ratio of the circumference of a circle to its diameter. static function IEEEremainder( f1 : Float, f2 : Float ) : FloatComputes the remainder operation on two arguments as prescribed by the IEEE 754 standard. The remainder value is mathematically equal to

f1 - f2 × <i>n</i>, where <i>n</i> is the mathematical integer closest to the exact mathematical value of the quotient {@code f1/f2}, and if two mathematical integers are equally close to {@code f1/f2}, then <i>n</i> is the integer that is even. If the remainder is zero, its sign is the same as the sign of the first argument. Special cases: <ul><li>If either argument is NaN, or the first argument is infinite, or the second argument is positive zero or negative zero, then the result is NaN. <li>If the first argument is finite and the second argument is infinite, then the result is the same as the first argument.</ul>

param f2 the divisor. @return the remainder when {@code f1} is divided by {@code f2}. static function abs( a : Int ) : IntReturns the absolute value of an {@code int} value. If the argument is not negative, the argument is returned. If the argument is negative, the negation of the argument is returned.

<p>Note that if the argument is equal to the value of {@link Integer#MIN_VALUE}, the most negative representable {@code int} value, the result is that same value, which is negative.

return the absolute value of the argument. static function acos( a : Float ) : FloatReturns the arc cosine of a value; the returned angle is in the range 0.0 through <i>pi</i>. Special case: <ul><li>If the argument is NaN or its absolute value is greater than 1, then the result is NaN.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the arc cosine of the argument. static function asin( a : Float ) : FloatReturns the arc sine of a value; the returned angle is in the range -<i>pi</i>/2 through <i>pi</i>/2. Special cases: <ul><li>If the argument is NaN or its absolute value is greater than 1, then the result is NaN. <li>If the argument is zero, then the result is a zero with the same sign as the argument.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the arc sine of the argument. static function atan( a : Float ) : FloatReturns the arc tangent of a value; the returned angle is in the range -<i>pi</i>/2 through <i>pi</i>/2. Special cases: <ul><li>If the argument is NaN, then the result is NaN. <li>If the argument is zero, then the result is a zero with the same sign as the argument.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the arc tangent of the argument. static function atan2( y : Float, x : Float ) : FloatReturns the angle <i>theta</i> from the conversion of rectangular coordinates ({@code x}, {@code y}) to polar coordinates (r, <i>theta</i>). This method computes the phase <i>theta</i> by computing an arc tangent of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special cases: <ul><li>If either argument is NaN, then the result is NaN. <li>If the first argument is positive zero and the second argument is positive, or the first argument is positive and finite and the second argument is positive infinity, then the result is positive zero. <li>If the first argument is negative zero and the second argument is positive, or the first argument is negative and finite and the second argument is positive infinity, then the result is negative zero. <li>If the first argument is positive zero and the second argument is negative, or the first argument is positive and finite and the second argument is negative infinity, then the result is the {@code double} value closest to <i>pi</i>. <li>If the first argument is negative zero and the second argument is negative, or the first argument is negative and finite and the second argument is negative infinity, then the result is the {@code double} value closest to -<i>pi</i>. <li>If the first argument is positive and the second argument is positive zero or negative zero, or the first argument is positive infinity and the second argument is finite, then the result is the {@code double} value closest to <i>pi</i>/2. <li>If the first argument is negative and the second argument is positive zero or negative zero, or the first argument is negative infinity and the second argument is finite, then the result is the {@code double} value closest to -<i>pi</i>/2. <li>If both arguments are positive infinity, then the result is the {@code double} value closest to <i>pi</i>/4. <li>If the first argument is positive infinity and the second argument is negative infinity, then the result is the {@code double} value closest to 3*<i>pi</i>/4. <li>If the first argument is negative infinity and the second argument is positive infinity, then the result is the {@code double} value closest to -<i>pi</i>/4. <li>If both arguments are negative infinity, then the result is the {@code double} value closest to -3*<i>pi</i>/4.</ul>

<p>The computed result must be within 2 ulps of the exact result. Results must be semi-monotonic.

param x the abscissa coordinate @return the <i>theta</i> component of the point (<i>r</i>, <i>theta</i>) in polar coordinates that corresponds to the point (<i>x</i>, <i>y</i>) in Cartesian coordinates. static function cbrt( a : Float ) : FloatReturns the cube root of a {@code double} value. For positive finite {@code x}, {@code cbrt(-x) == -cbrt(x)}; that is, the cube root of a negative value is the negative of the cube root of that value's magnitude.

Special cases:

<ul>

<li>If the argument is NaN, then the result is NaN.

<li>If the argument is infinite, then the result is an infinity with the same sign as the argument.

<li>If the argument is zero, then the result is a zero with the same sign as the argument.

</ul>

<p>The computed result must be within 1 ulp of the exact result.

return the cube root of {@code a}. @since 1.5 static function ceil( a : Float ) : FloatReturns the smallest (closest to negative infinity) {@code double} value that is greater than or equal to the argument and is equal to a mathematical integer. Special cases: <ul><li>If the argument value is already equal to a mathematical integer, then the result is the same as the argument. <li>If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument. <li>If the argument value is less than zero but greater than -1.0, then the result is negative zero.</ul> Note that the value of {@code Math.ceil(x)} is exactly the value of {@code -Math.floor(-x)}.

return the smallest (closest to negative infinity) floating-point value that is greater than or equal to the argument and is equal to a mathematical integer. static function copySign( magnitude : Float, sign : Float ) : FloatReturns the first floating-point argument with the sign of the second floating-point argument. Note that unlike the {@link StrictMath#copySign(double, double) StrictMath.copySign} method, this method does not require NaN {@code sign} arguments to be treated as positive values; implementations are permitted to treat some NaN arguments as positive and other NaN arguments as negative to allow greater performance.

param sign the parameter providing the sign of the result @return a value with the magnitude of {@code magnitude} and the sign of {@code sign}. @since 1.6 static function cos( a : Float ) : FloatReturns the trigonometric cosine of an angle. Special cases: <ul><li>If the argument is NaN or an infinity, then the result is NaN.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

@param a an angle, in radians. @return the cosine of the argument. static function cosh( x : Float ) : FloatReturns the hyperbolic cosine of a {@code double} value. The hyperbolic cosine of <i>x</i> is defined to be (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2 where <i>e</i> is {@linkplain Math#E Euler's number}.

<p>Special cases: <ul>

<li>If the argument is NaN, then the result is NaN.

<li>If the argument is infinite, then the result is positive infinity.

<li>If the argument is zero, then the result is {@code 1.0}.

</ul>

<p>The computed result must be within 2.5 ulps of the exact result.

return The hyperbolic cosine of {@code x}. @since 1.5 static function exp( a : Float ) : FloatReturns Euler's number <i>e</i> raised to the power of a {@code double} value. Special cases: <ul><li>If the argument is NaN, the result is NaN. <li>If the argument is positive infinity, then the result is positive infinity. <li>If the argument is negative infinity, then the result is positive zero.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

@param a the exponent to raise <i>e</i> to. @return the value <i>e</i><sup>{@code a}</sup>, where <i>e</i> is the base of the natural logarithms. static function expm1( x : Float ) : FloatReturns <i>e</i><sup>x</sup> -1. Note that for values of <i>x</i> near 0, the exact sum of {@code expm1(x)} + 1 is much closer to the true result of <i>e</i><sup>x</sup> than {@code exp(x)}.

<p>Special cases: <ul> <li>If the argument is NaN, the result is NaN.

<li>If the argument is positive infinity, then the result is positive infinity.

<li>If the argument is negative infinity, then the result is -1.0.

<li>If the argument is zero, then the result is a zero with the same sign as the argument.

</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic. The result of {@code expm1} for any finite input must be greater than or equal to {@code -1.0}. Note that once the exact result of <i>e</i><sup>{@code x}</sup> - 1 is within 1/2 ulp of the limit value -1, {@code -1.0} should be returned.

@param x the exponent to raise <i>e</i> to in the computation of <i>e</i><sup>{@code x}</sup> -1. @return the value <i>e</i><sup>{@code x}</sup> - 1. @since 1.5 static function floor( a : Float ) : FloatReturns the largest (closest to positive infinity) {@code double} value that is less than or equal to the argument and is equal to a mathematical integer. Special cases: <ul><li>If the argument value is already equal to a mathematical integer, then the result is the same as the argument. <li>If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument.</ul>

return the largest (closest to positive infinity) floating-point value that less than or equal to the argument and is equal to a mathematical integer. static function getExponent( f : Single ) : IntReturns the unbiased exponent used in the representation of a {@code float}. Special cases:

<ul> <li>If the argument is NaN or infinite, then the result is {@link Float#MAX_EXPONENT} + 1. <li>If the argument is zero or subnormal, then the result is {@link Float#MIN_EXPONENT} -1. </ul> @param f a {@code float} value since 1.6 static function hypot( x : Float, y : Float ) : FloatReturns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) without intermediate overflow or underflow.

<p>Special cases: <ul>

<li> If either argument is infinite, then the result is positive infinity.

<li> If either argument is NaN and neither argument is infinite, then the result is NaN.

</ul>

<p>The computed result must be within 1 ulp of the exact result. If one parameter is held constant, the results must be semi-monotonic in the other parameter.

param y a value @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) without intermediate overflow or underflow @since 1.5 static function log( a : Float ) : FloatReturns the natural logarithm (base <i>e</i>) of a {@code double} value. Special cases: <ul><li>If the argument is NaN or less than zero, then the result is NaN. <li>If the argument is positive infinity, then the result is positive infinity. <li>If the argument is positive zero or negative zero, then the result is negative infinity.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the value ln {@code a}, the natural logarithm of {@code a}. static function log10( a : Float ) : FloatReturns the base 10 logarithm of a {@code double} value. Special cases:

<ul><li>If the argument is NaN or less than zero, then the result is NaN. <li>If the argument is positive infinity, then the result is positive infinity. <li>If the argument is positive zero or negative zero, then the result is negative infinity. <li> If the argument is equal to 10<sup><i>n</i></sup> for integer <i>n</i>, then the result is <i>n</i>. </ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the base 10 logarithm of {@code a}. @since 1.5 static function log1p( x : Float ) : FloatReturns the natural logarithm of the sum of the argument and 1. Note that for small values {@code x}, the result of {@code log1p(x)} is much closer to the true result of ln(1 + {@code x}) than the floating-point evaluation of {@code log(1.0+x)}.

<p>Special cases:

<ul>

<li>If the argument is NaN or less than -1, then the result is NaN.

<li>If the argument is positive infinity, then the result is positive infinity.

<li>If the argument is negative one, then the result is negative infinity.

<li>If the argument is zero, then the result is a zero with the same sign as the argument.

</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

return the value ln({@code x} + 1), the natural log of {@code x} + 1 @since 1.5 static function max( a : Int, b : Int ) : IntReturns the greater of two {@code int} values. That is, the result is the argument closer to the value of {@link Integer#MAX_VALUE}. If the arguments have the same value, the result is that same value.

param b another argument. @return the larger of {@code a} and {@code b}. static function min( a : Int, b : Int ) : IntReturns the smaller of two {@code int} values. That is, the result the argument closer to the value of {@link Integer#MIN_VALUE}. If the arguments have the same value, the result is that same value.

param b another argument. @return the smaller of {@code a} and {@code b}. static function nextAfter( start : Float, direction : Float ) : FloatReturns the floating-point number adjacent to the first argument in the direction of the second argument. If both arguments compare as equal the second argument is returned.

<p> Special cases: <ul> <li> If either argument is a NaN, then NaN is returned.

<li> If both arguments are signed zeros, {@code direction} is returned unchanged (as implied by the requirement of returning the second argument if the arguments compare as equal).

<li> If {@code start} is ±{@link Double#MIN_VALUE} and {@code direction} has a value such that the result should have a smaller magnitude, then a zero with the same sign as {@code start} is returned.

<li> If {@code start} is infinite and {@code direction} has a value such that the result should have a smaller magnitude, {@link Double#MAX_VALUE} with the same sign as {@code start} is returned.

<li> If {@code start} is equal to ± {@link Double#MAX_VALUE} and {@code direction} has a value such that the result should have a larger magnitude, an infinity with same sign as {@code start} is returned. </ul>

param direction value indicating which of {@code start}'s neighbors or {@code start} should be returned @return The floating-point number adjacent to {@code start} in the direction of {@code direction}. @since 1.6 static function nextUp( d : Float ) : FloatReturns the floating-point value adjacent to {@code d} in the direction of positive infinity. This method is semantically equivalent to {@code nextAfter(d, Double.POSITIVE_INFINITY)}; however, a {@code nextUp} implementation may run faster than its equivalent {@code nextAfter} call.

<p>Special Cases: <ul> <li> If the argument is NaN, the result is NaN.

<li> If the argument is positive infinity, the result is positive infinity.

<li> If the argument is zero, the result is {@link Double#MIN_VALUE}

</ul>

return The adjacent floating-point value closer to positive infinity. @since 1.6 static function pow( a : Float, b : Float ) : FloatReturns the value of the first argument raised to the power of the second argument. Special cases:

<ul><li>If the second argument is positive or negative zero, then the result is 1.0. <li>If the second argument is 1.0, then the result is the same as the first argument. <li>If the second argument is NaN, then the result is NaN. <li>If the first argument is NaN and the second argument is nonzero, then the result is NaN.

<li>If <ul> <li>the absolute value of the first argument is greater than 1 and the second argument is positive infinity, or <li>the absolute value of the first argument is less than 1 and the second argument is negative infinity, </ul> then the result is positive infinity.

<li>If <ul> <li>the absolute value of the first argument is greater than 1 and the second argument is negative infinity, or <li>the absolute value of the first argument is less than 1 and the second argument is positive infinity, </ul> then the result is positive zero.

<li>If the absolute value of the first argument equals 1 and the second argument is infinite, then the result is NaN.

<li>If <ul> <li>the first argument is positive zero and the second argument is greater than zero, or <li>the first argument is positive infinity and the second argument is less than zero, </ul> then the result is positive zero.

<li>If <ul> <li>the first argument is positive zero and the second argument is less than zero, or <li>the first argument is positive infinity and the second argument is greater than zero, </ul> then the result is positive infinity.

<li>If <ul> <li>the first argument is negative zero and the second argument is greater than zero but not a finite odd integer, or <li>the first argument is negative infinity and the second argument is less than zero but not a finite odd integer, </ul> then the result is positive zero.

<li>If <ul> <li>the first argument is negative zero and the second argument is a positive finite odd integer, or <li>the first argument is negative infinity and the second argument is a negative finite odd integer, </ul> then the result is negative zero.

<li>If <ul> <li>the first argument is negative zero and the second argument is less than zero but not a finite odd integer, or <li>the first argument is negative infinity and the second argument is greater than zero but not a finite odd integer, </ul> then the result is positive infinity.

<li>If <ul> <li>the first argument is negative zero and the second argument is a negative finite odd integer, or <li>the first argument is negative infinity and the second argument is a positive finite odd integer, </ul> then the result is negative infinity.

<li>If the first argument is finite and less than zero <ul> <li> if the second argument is a finite even integer, the result is equal to the result of raising the absolute value of the first argument to the power of the second argument

<li>if the second argument is a finite odd integer, the result is equal to the negative of the result of raising the absolute value of the first argument to the power of the second argument

<li>if the second argument is finite and not an integer, then the result is NaN. </ul>

<li>If both arguments are integers, then the result is exactly equal to the mathematical result of raising the first argument to the power of the second argument if that result can in fact be represented exactly as a {@code double} value.</ul>

<p>(In the foregoing descriptions, a floating-point value is considered to be an integer if and only if it is finite and a fixed point of the method {@link #ceil ceil} or, equivalently, a fixed point of the method {@link #floor floor}. A value is a fixed point of a one-argument method if and only if the result of applying the method to the value is equal to the value.)

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

param b the exponent. @return the value {@code a}<sup>{@code b}</sup>. static function random() : FloatReturns a {@code double} value with a positive sign, greater than or equal to {@code 0.0} and less than {@code 1.0}. Returned values are chosen pseudorandomly with (approximately) uniform distribution from that range.

<p>When this method is first called, it creates a single new pseudorandom-number generator, exactly as if by the expression

<blockquote>{@code new java.util.Random()}</blockquote>

This new pseudorandom-number generator is used thereafter for all calls to this method and is used nowhere else.

<p>This method is properly synchronized to allow correct use by more than one thread. However, if many threads need to generate pseudorandom numbers at a great rate, it may reduce contention for each thread to have its own pseudorandom-number generator.

@return a pseudorandom {@code double} greater than or equal to {@code 0.0} and less than {@code 1.0}. @see Random#nextDouble() static function rint( a : Float ) : FloatReturns the {@code double} value that is closest in value to the argument and is equal to a mathematical integer. If two {@code double} values that are mathematical integers are equally close, the result is the integer value that is even. Special cases: <ul><li>If the argument value is already equal to a mathematical integer, then the result is the same as the argument. <li>If the argument is NaN or an infinity or positive zero or negative zero, then the result is the same as the argument.</ul>

@param a a {@code double} value. @return the closest floating-point value to {@code a} that is equal to a mathematical integer. static function round( a : Single ) : IntReturns the closest {@code int} to the argument, with ties rounding up.

<p> Special cases: <ul><li>If the argument is NaN, the result is 0. <li>If the argument is negative infinity or any value less than or equal to the value of {@code Integer.MIN_VALUE}, the result is equal to the value of {@code Integer.MIN_VALUE}. <li>If the argument is positive infinity or any value greater than or equal to the value of {@code Integer.MAX_VALUE}, the result is equal to the value of {@code Integer.MAX_VALUE}.</ul>

return the value of the argument rounded to the nearest {@code int} value. @see java.lang.Integer#MAX_VALUE @see java.lang.Integer#MIN_VALUE static function scalb( d : Float, scaleFactor : Int ) : FloatReturn {@code d} × 2<sup>{@code scaleFactor}</sup> rounded as if performed by a single correctly rounded floating-point multiply to a member of the double value set. See the Java Language Specification for a discussion of floating-point value sets. If the exponent of the result is between {@link Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the answer is calculated exactly. If the exponent of the result would be larger than {@code Double.MAX_EXPONENT}, an infinity is returned. Note that if the result is subnormal, precision may be lost; that is, when {@code scalb(x, n)} is subnormal, {@code scalb(scalb(x, n), -n)} may not equal <i>x</i>. When the result is non-NaN, the result has the same sign as {@code d}.

<p>Special cases: <ul> <li> If the first argument is NaN, NaN is returned. <li> If the first argument is infinite, then an infinity of the same sign is returned. <li> If the first argument is zero, then a zero of the same sign is returned. </ul>

param scaleFactor power of 2 used to scale {@code d} @return {@code d} × 2<sup>{@code scaleFactor}</sup> @since 1.6 static function signum( d : Float ) : FloatReturns the signum function of the argument; zero if the argument is zero, 1.0 if the argument is greater than zero, -1.0 if the argument is less than zero.

<p>Special Cases: <ul> <li> If the argument is NaN, then the result is NaN. <li> If the argument is positive zero or negative zero, then the result is the same as the argument. </ul>

return the signum function of the argument since 1.5 static function sin( a : Float ) : FloatReturns the trigonometric sine of an angle. Special cases: <ul><li>If the argument is NaN or an infinity, then the result is NaN. <li>If the argument is zero, then the result is a zero with the same sign as the argument.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

@param a an angle, in radians. @return the sine of the argument. static function sinh( x : Float ) : FloatReturns the hyperbolic sine of a {@code double} value. The hyperbolic sine of <i>x</i> is defined to be (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2 where <i>e</i> is {@linkplain Math#E Euler's number}.

<p>Special cases: <ul>

<li>If the argument is NaN, then the result is NaN.

<li>If the argument is infinite, then the result is an infinity with the same sign as the argument.

<li>If the argument is zero, then the result is a zero with the same sign as the argument.

</ul>

<p>The computed result must be within 2.5 ulps of the exact result.

return The hyperbolic sine of {@code x}. @since 1.5 static function sqrt( a : Float ) : FloatReturns the correctly rounded positive square root of a {@code double} value. Special cases: <ul><li>If the argument is NaN or less than zero, then the result is NaN. <li>If the argument is positive infinity, then the result is positive infinity. <li>If the argument is positive zero or negative zero, then the result is the same as the argument.</ul> Otherwise, the result is the {@code double} value closest to the true mathematical square root of the argument value.

return the positive square root of {@code a}. If the argument is NaN or less than zero, the result is NaN. static function tan( a : Float ) : FloatReturns the trigonometric tangent of an angle. Special cases: <ul><li>If the argument is NaN or an infinity, then the result is NaN. <li>If the argument is zero, then the result is a zero with the same sign as the argument.</ul>

<p>The computed result must be within 1 ulp of the exact result. Results must be semi-monotonic.

@param a an angle, in radians. @return the tangent of the argument. static function tanh( x : Float ) : FloatReturns the hyperbolic tangent of a {@code double} value. The hyperbolic tangent of <i>x</i> is defined to be (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>), in other words, {@linkplain Math#sinh sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note that the absolute value of the exact tanh is always less than 1.

<p>Special cases: <ul>

<li>If the argument is NaN, then the result is NaN.

<li>If the argument is zero, then the result is a zero with the same sign as the argument.

<li>If the argument is positive infinity, then the result is {@code +1.0}.

<li>If the argument is negative infinity, then the result is {@code -1.0}.

</ul>

<p>The computed result must be within 2.5 ulps of the exact result. The result of {@code tanh} for any finite input must have an absolute value less than or equal to 1. Note that once the exact result of tanh is within 1/2 of an ulp of the limit value of ±1, correctly signed ±{@code 1.0} should be returned.

return The hyperbolic tangent of {@code x}. @since 1.5 static function toDegrees( angrad : Float ) : FloatConverts an angle measured in radians to an approximately equivalent angle measured in degrees. The conversion from radians to degrees is generally inexact; users should <i>not</i> expect {@code cos(toRadians(90.0))} to exactly equal {@code 0.0}.

@param angrad an angle, in radians @return the measurement of the angle {@code angrad} in degrees. @since 1.2 static function toRadians( angdeg : Float ) : FloatConverts an angle measured in degrees to an approximately equivalent angle measured in radians. The conversion from degrees to radians is generally inexact.

@param angdeg an angle, in degrees @return the measurement of the angle {@code angdeg} in radians. @since 1.2 static function ulp( d : Float ) : FloatReturns the size of an ulp of the argument. An ulp of a {@code double} value is the positive distance between this floating-point value and the {@code double} value next larger in magnitude. Note that for non-NaN <i>x</i>,

ulp(-<i>x</i>) == ulp(<i>x</i>).

<p>Special Cases: <ul> <li> If the argument is NaN, then the result is NaN. <li> If the argument is positive or negative infinity, then the result is positive infinity. <li> If the argument is positive or negative zero, then the result is {@code Double.MIN_VALUE}. <li> If the argument is ±{@code Double.MAX_VALUE}, then the result is equal to 2<sup>971</sup>. </ul>

return the size of an ulp of the argument since 1.5

version #16398, modified 2013-02-24 23:52:19 by api

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