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## org.apache.commons.math.util
Class FastMath

    java.lang.Object
      org.apache.commons.math.util.FastMath

    public class FastMath

extends Object

Faster, more accurate, portable alternative to `StrictMath`.

Additionally implements the following methods not found in StrictMath:

- `asinh(double)`
- `acosh(double)`
- `atanh(double)`

The following methods are found in StrictMath since 1.6 only

- `copySign(double, double)`
- `getExponent(double)`
- `nextAfter(double,double)`
- `nextUp(double)`
- `scalb(double, int)`
- `copySign(float, float)`
- `getExponent(float)`
- `nextAfter(float,double)`
- `nextUp(float)`
- `scalb(float, int)`

Since:
  2.2
Version:
  $Revision: 1074294 $ $Date: 2011-02-24 22:18:59 +0100 (jeu. 24 févr. 2011) $

Field Summary
`private static double[]` | `CBRTTWO` Table of 2^((n+2)/3)
`private static double[]` | `COSINE_TABLE_A` Cosine table (high bits).
`private static double[]` | `COSINE_TABLE_B` Cosine table (low bits).
`static double` | `E` Napier's constant e, base of the natural logarithm.
`private static double[]` | `EIGHTHS` Eighths.
`private static double[]` | `EXP_FRAC_TABLE_A` Exponential over the range of 0 - 1 in increments of 2^-10 exp(x/1024) = expFracTableA[x] + expFracTableB[x].
`private static double[]` | `EXP_FRAC_TABLE_B` Exponential over the range of 0 - 1 in increments of 2^-10 exp(x/1024) = expFracTableA[x] + expFracTableB[x].
`private static double[]` | `EXP_INT_TABLE_A` Exponential evaluated at integer values, exp(x) = expIntTableA[x + 750] + expIntTableB[x+750].
`private static double[]` | `EXP_INT_TABLE_B` Exponential evaluated at integer values, exp(x) = expIntTableA[x + 750] + expIntTableB[x+750]
`private static double[]` | `FACT` Factorial table, for Taylor series expansions.
`private static long` | `HEX_40000000` 0x40000000 - used to split a double into two parts, both with the low order bits cleared.
`private static double` | `LN_2_A` log(2) (high bits).
`private static double` | `LN_2_B` log(2) (low bits).
`private static double[][]` | `LN_HI_PREC_COEF` Coefficients for log in the range of 1.0 < x < 1.0 + 2^-10.
`private static double[][]` | `LN_MANT` Extended precision logarithm table over the range 1 - 2 in increments of 2^-10.
`private static double[][]` | `LN_QUICK_COEF` Coefficients for log, when input 0.99 < x < 1.01.
`private static double[][]` | `LN_SPLIT_COEF` Coefficients for slowLog.
`private static long` | `MASK_30BITS` Mask used to clear low order 30 bits
`static double` | `PI` Archimede's constant PI, ratio of circle circumference to diameter.
`private static long[]` | `PI_O_4_BITS` Bits of pi/4, need for reducePayneHanek().
`private static long[]` | `RECIP_2PI` Bits of 1/(2*pi), need for reducePayneHanek().
`private static double[]` | `SINE_TABLE_A` Sine table (high bits).
`private static double[]` | `SINE_TABLE_B` Sine table (low bits).
`private static double[]` | `TANGENT_TABLE_A` Tangent table, used by atan() (high bits).
`private static double[]` | `TANGENT_TABLE_B` Tangent table, used by atan() (low bits).
`private static double` | `TWO_POWER_52` 2^52 - double numbers this large must be integral (no fraction) or NaN or Infinite

Constructor Summary
`private` | `FastMath()` Private Constructor

Method Summary
`static double` | `abs(double x)` Absolute value.
`static float` | `abs(float x)` Absolute value.
`static int` | `abs(int x)` Absolute value.
`static long` | `abs(long x)` Absolute value.
`static double` | `acos(double x)` Compute the arc cosine of a number.
`static double` | `acosh(double a)` Compute the inverse hyperbolic cosine of a number.
`static double` | `asin(double x)` Compute the arc sine of a number.
`static double` | `asinh(double a)` Compute the inverse hyperbolic sine of a number.
`static double` | `atan(double x)` Arctangent function
`private static double` | `atan(double xa, double xb, boolean leftPlane)` Internal helper function to compute arctangent.
`static double` | `atan2(double y, double x)` Two arguments arctangent function
`static double` | `atanh(double a)` Compute the inverse hyperbolic tangent of a number.
`private static void` | `buildSinCosTables()` Build the sine and cosine tables.
`static double` | `cbrt(double x)` Compute the cubic root of a number.
`static double` | `ceil(double x)` Get the smallest whole number larger than x.
`static double` | `copySign(double magnitude, double sign)` Returns the first argument with the sign of the second argument.
`static float` | `copySign(float magnitude, float sign)` Returns the first argument with the sign of the second argument.
`static double` | `cos(double x)` Cosine function
`static double` | `cosh(double x)` Compute the hyperbolic cosine of a number.
`private static double` | `cosQ(double xa, double xb)` Compute cosine in the first quadrant by subtracting input from PI/2 and then calling sinQ.
`private static double` | `doubleHighPart(double d)` Get the high order bits from the mantissa.
`static double` | `exp(double x)` Exponential function.
`private static double` | `exp(double x, double extra, double[] hiPrec)` Internal helper method for exponential function.
`private static double` | `expint(int p, double[] result)` Compute exp(p) for a integer p in extended precision.
`static double` | `expm1(double x)` Compute exp(x) - 1
`private static double` | `expm1(double x, double[] hiPrecOut)` Internal helper method for expm1
`static double` | `floor(double x)` Get the largest whole number smaller than x.
`static int` | `getExponent(double d)` Return the exponent of a double number, removing the bias.
`static int` | `getExponent(float f)` Return the exponent of a float number, removing the bias.
`static double` | `hypot(double x, double y)` Returns the hypotenuse of a triangle with sides `x` and `y` - sqrt(x2 +y2) avoiding intermediate overflow or underflow.
`static double` | `IEEEremainder(double dividend, double divisor)` Computes the remainder as prescribed by the IEEE 754 standard.
`static double` | `log(double x)` Natural logarithm.
`private static double` | `log(double x, double[] hiPrec)` Internal helper method for natural logarithm function.
`static double` | `log10(double x)` Compute the base 10 logarithm.
`static double` | `log1p(double x)` Compute log(1 + x).
`static double` | `max(double a, double b)` Compute the maximum of two values
`static float` | `max(float a, float b)` Compute the maximum of two values
`static int` | `max(int a, int b)` Compute the maximum of two values
`static long` | `max(long a, long b)` Compute the maximum of two values
`static double` | `min(double a, double b)` Compute the minimum of two values
`static float` | `min(float a, float b)` Compute the minimum of two values
`static int` | `min(int a, int b)` Compute the minimum of two values
`static long` | `min(long a, long b)` Compute the minimum of two values
`static double` | `nextAfter(double d, double direction)` Get the next machine representable number after a number, moving in the direction of another number.
`static float` | `nextAfter(float f, double direction)` Get the next machine representable number after a number, moving in the direction of another number.
`static double` | `nextUp(double a)` Compute next number towards positive infinity.
`static float` | `nextUp(float a)` Compute next number towards positive infinity.
`private static double` | `polyCosine(double x)` Computes cos(x) - 1, where |x| < 1/16.
`private static double` | `polySine(double x)` Computes sin(x) - x, where |x| < 1/16.
`static double` | `pow(double x, double y)` Power function.
`private static void` | `quadMult(double[] a, double[] b, double[] result)` Compute (a[0] + a[1]) * (b[0] + b[1]) in extended precision.
`static double` | `random()` Returns a pseudo-random number between 0.0 and 1.0.
`private static void` | `reducePayneHanek(double x, double[] result)` Reduce the input argument using the Payne and Hanek method.
`private static void` | `resplit(double[] a)` Recompute a split.
`static double` | `rint(double x)` Get the whole number that is the nearest to x, or the even one if x is exactly half way between two integers.
`static long` | `round(double x)` Get the closest long to x.
`static int` | `round(float x)` Get the closest int to x.
`static double` | `scalb(double d, int n)` Multiply a double number by a power of 2.
`static float` | `scalb(float f, int n)` Multiply a float number by a power of 2.
`static double` | `signum(double a)` Compute the signum of a number.
`static float` | `signum(float a)` Compute the signum of a number.
`static double` | `sin(double x)` Sine function.
`static double` | `sinh(double x)` Compute the hyperbolic sine of a number.
`private static double` | `sinQ(double xa, double xb)` Compute sine over the first quadrant (0 < x < pi/2).
`private static double` | `slowCos(double x, double[] result)` For x between 0 and pi/4 compute cosine
`private static double` | `slowexp(double x, double[] result)` For x between 0 and 1, returns exp(x), uses extended precision
`private static double[]` | `slowLog(double xi)` xi in the range of [1, 2].
`private static double` | `slowSin(double x, double[] result)` For x between 0 and pi/4 compute sine.
`private static void` | `split(double d, double[] split)` Compute split[0], split[1] such that their sum is equal to d, and split[0] has its 30 least significant bits as zero.
`private static void` | `splitAdd(double[] a, double[] b, double[] ans)` Add two numbers in split form.
`private static void` | `splitMult(double[] a, double[] b, double[] ans)` Multiply two numbers in split form.
`private static void` | `splitReciprocal(double[] in, double[] result)` Compute the reciprocal of in.
`static double` | `sqrt(double a)` Compute the square root of a number.
`static double` | `tan(double x)` Tangent function
`static double` | `tanh(double x)` Compute the hyperbolic tangent of a number.
`private static double` | `tanQ(double xa, double xb, boolean cotanFlag)` Compute tangent (or cotangent) over the first quadrant.
`static double` | `toDegrees(double x)` Convert radians to degrees, with error of less than 0.5 ULP
`static double` | `toRadians(double x)` Convert degrees to radians, with error of less than 0.5 ULP
`static double` | `ulp(double x)` Compute least significant bit (Unit in Last Position) for a number.
`static float` | `ulp(float x)` Compute least significant bit (Unit in Last Position) for a number.

Methods inherited from class java.lang.Object
`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

Field Detail

### PI

    public static final double PI

  Archimede's constant PI, ratio of circle circumference to diameter.

  See Also:
    Constant Field Values

### E

    public static final double E

  Napier's constant e, base of the natural logarithm.

  See Also:
    Constant Field Values

### EXP_INT_TABLE_A

    private static final double[] EXP_INT_TABLE_A

  Exponential evaluated at integer values, exp(x) = expIntTableA[x + 750] + expIntTableB[x+750].

### EXP_INT_TABLE_B

    private static final double[] EXP_INT_TABLE_B

  Exponential evaluated at integer values, exp(x) = expIntTableA[x + 750] + expIntTableB[x+750]

### EXP_FRAC_TABLE_A

    private static final double[] EXP_FRAC_TABLE_A

  Exponential over the range of 0 - 1 in increments of 2^-10 exp(x/1024) = expFracTableA[x] + expFracTableB[x].

### EXP_FRAC_TABLE_B

    private static final double[] EXP_FRAC_TABLE_B

  Exponential over the range of 0 - 1 in increments of 2^-10 exp(x/1024) = expFracTableA[x] + expFracTableB[x].

### FACT

    private static final double[] FACT

  Factorial table, for Taylor series expansions.

### LN_MANT

    private static final double[][] LN_MANT

  Extended precision logarithm table over the range 1 - 2 in increments of 2^-10.

### LN_2_A

    private static final double LN_2_A

  log(2) (high bits).

  See Also:
    Constant Field Values

### LN_2_B

    private static final double LN_2_B

  log(2) (low bits).

  See Also:
    Constant Field Values

### LN_SPLIT_COEF

    private static final double[][] LN_SPLIT_COEF

  Coefficients for slowLog.

### LN_QUICK_COEF

    private static final double[][] LN_QUICK_COEF

  Coefficients for log, when input 0.99 < x < 1.01.

### LN_HI_PREC_COEF

    private static final double[][] LN_HI_PREC_COEF

  Coefficients for log in the range of 1.0 < x < 1.0 + 2^-10.

### SINE_TABLE_A

    private static final double[] SINE_TABLE_A

  Sine table (high bits).

### SINE_TABLE_B

    private static final double[] SINE_TABLE_B

  Sine table (low bits).

### COSINE_TABLE_A

    private static final double[] COSINE_TABLE_A

  Cosine table (high bits).

### COSINE_TABLE_B

    private static final double[] COSINE_TABLE_B

  Cosine table (low bits).

### TANGENT_TABLE_A

    private static final double[] TANGENT_TABLE_A

  Tangent table, used by atan() (high bits).

### TANGENT_TABLE_B

    private static final double[] TANGENT_TABLE_B

  Tangent table, used by atan() (low bits).

### RECIP_2PI

    private static final long[] RECIP_2PI

  Bits of 1/(2*pi), need for reducePayneHanek().

### PI_O_4_BITS

    private static final long[] PI_O_4_BITS

  Bits of pi/4, need for reducePayneHanek().

### EIGHTHS

    private static final double[] EIGHTHS

  Eighths. This is used by sinQ, because its faster to do a table lookup than a multiply in this time-critical routine

### CBRTTWO

    private static final double[] CBRTTWO

  Table of 2^((n+2)/3)

### HEX_40000000

    private static final long HEX_40000000

  0x40000000 - used to split a double into two parts, both with the low order bits cleared. Equivalent to 2^30.

  See Also:
    Constant Field Values

### MASK_30BITS

    private static final long MASK_30BITS

  Mask used to clear low order 30 bits

  See Also:
    Constant Field Values

### TWO_POWER_52

    private static final double TWO_POWER_52

  2^52 - double numbers this large must be integral (no fraction) or NaN or Infinite

  See Also:
    Constant Field Values

Constructor Detail

### FastMath

    private FastMath()

  Private Constructor

Method Detail

### doubleHighPart

    private static double doubleHighPart(double d)

  Get the high order bits from the mantissa. Equivalent to adding and subtracting HEX_40000 but also works for very large numbers

  Parameters:
    `d` - the value to split
  Returns:
    the high order part of the mantissa

### sqrt

    public static double sqrt(double a)

  Compute the square root of a number.

  Note: this implementation currently delegates to `Math.sqrt(double)`

  Parameters:
    `a` - number on which evaluation is done
  Returns:
    square root of a

### cosh

    public static double cosh(double x)

  Compute the hyperbolic cosine of a number.

  Parameters:
    `x` - number on which evaluation is done
  Returns:
    hyperbolic cosine of x

### sinh

    public static double sinh(double x)

  Compute the hyperbolic sine of a number.

  Parameters:
    `x` - number on which evaluation is done
  Returns:
    hyperbolic sine of x

### tanh

    public static double tanh(double x)

  Compute the hyperbolic tangent of a number.

  Parameters:
    `x` - number on which evaluation is done
  Returns:
    hyperbolic tangent of x

### acosh

    public static double acosh(double a)

  Compute the inverse hyperbolic cosine of a number.

  Parameters:
    `a` - number on which evaluation is done
  Returns:
    inverse hyperbolic cosine of a

### asinh

    public static double asinh(double a)

  Compute the inverse hyperbolic sine of a number.

  Parameters:
    `a` - number on which evaluation is done
  Returns:
    inverse hyperbolic sine of a

### atanh

    public static double atanh(double a)

  Compute the inverse hyperbolic tangent of a number.

  Parameters:
    `a` - number on which evaluation is done
  Returns:
    inverse hyperbolic tangent of a

### signum

    public static double signum(double a)

  Compute the signum of a number. The signum is -1 for negative numbers, +1 for positive numbers and 0 otherwise

  Parameters:
    `a` - number on which evaluation is done
  Returns:
    -1.0, -0.0, +0.0, +1.0 or NaN depending on sign of a

### signum

    public static float signum(float a)

  Compute the signum of a number. The signum is -1 for negative numbers, +1 for positive numbers and 0 otherwise

  Parameters:
    `a` - number on which evaluation is done
  Returns:
    -1.0, -0.0, +0.0, +1.0 or NaN depending on sign of a

### nextUp

    public static double nextUp(double a)

  Compute next number towards positive infinity.

  Parameters:
    `a` - number to which neighbor should be computed
  Returns:
    neighbor of a towards positive infinity

### nextUp

    public static float nextUp(float a)

  Compute next number towards positive infinity.

  Parameters:
    `a` - number to which neighbor should be computed
  Returns:
    neighbor of a towards positive infinity

### random

    public static double random()

  Returns a pseudo-random number between 0.0 and 1.0.

  Note: this implementation currently delegates to `Math.random()`

  Returns:
    a random number between 0.0 and 1.0

### exp

    public static double exp(double x)

  Exponential function. Computes exp(x), function result is nearly rounded. It will be correctly rounded to the theoretical value for 99.9% of input values, otherwise it will have a 1 UPL error. Method: Lookup intVal = exp(int(x)) Lookup fracVal = exp(int(x-int(x) / 1024.0) * 1024.0 ); Compute z as the exponential of the remaining bits by a polynomial minus one exp(x) = intVal * fracVal * (1 + z) Accuracy: Calculation is done with 63 bits of precision, so result should be correctly rounded for 99.9% of input values, with less than 1 ULP error otherwise.

  Parameters:
    `x` - a double
  Returns:
    double ex

### exp

    private static double exp(double x,
                              double extra,
                              double[] hiPrec)

  Internal helper method for exponential function.

  Parameters:
    `x` - original argument of the exponential function
    `extra` - extra bits of precision on input (To Be Confirmed)
    `hiPrec` - extra bits of precision on output (To Be Confirmed)
  Returns:
    exp(x)

### expm1

    public static double expm1(double x)

  Compute exp(x) - 1

  Parameters:
    `x` - number to compute shifted exponential
  Returns:
    exp(x) - 1

### expm1

    private static double expm1(double x,
                                double[] hiPrecOut)

  Internal helper method for expm1

  Parameters:
    `x` - number to compute shifted exponential
    `hiPrecOut` - receive high precision result for -1.0 < x < 1.0
  Returns:
    exp(x) - 1

### slowexp

    private static double slowexp(double x,
                                  double[] result)

  For x between 0 and 1, returns exp(x), uses extended precision

  Parameters:
    `x` - argument of exponential
    `result` - placeholder where to place exp(x) split in two terms for extra precision (i.e. exp(x) = result[0] ° result[1]
  Returns:
    exp(x)

### split

    private static void split(double d,
                              double[] split)

  Compute split[0], split[1] such that their sum is equal to d, and split[0] has its 30 least significant bits as zero.

  Parameters:
    `d` - number to split
    `split` - placeholder where to place the result

### resplit

    private static void resplit(double[] a)

  Recompute a split.

  Parameters:
    `a` - input/out array containing the split, changed on output

### splitMult

    private static void splitMult(double[] a,
                                  double[] b,
                                  double[] ans)

  Multiply two numbers in split form.

  Parameters:
    `a` - first term of multiplication
    `b` - second term of multiplication
    `ans` - placeholder where to put the result

### splitAdd

    private static void splitAdd(double[] a,
                                 double[] b,
                                 double[] ans)

  Add two numbers in split form.

  Parameters:
    `a` - first term of addition
    `b` - second term of addition
    `ans` - placeholder where to put the result

### splitReciprocal

    private static void splitReciprocal(double[] in,
                                        double[] result)

  Compute the reciprocal of in. Use the following algorithm. in = c + d. want to find x + y such that x+y = 1/(c+d) and x is much larger than y and x has several zero bits on the right. Set b = 1/(2^22), a = 1 - b. Thus (a+b) = 1. Use following identity to compute (a+b)/(c+d) (a+b)/(c+d) = a/c + (bc - ad) / (c^2 + cd) set x = a/c and y = (bc - ad) / (c^2 + cd) This will be close to the right answer, but there will be some rounding in the calculation of X. So by carefully computing 1 - (c+d)(x+y) we can compute an error and add that back in. This is done carefully so that terms of similar size are subtracted first.

  Parameters:
    `in` - initial number, in split form
    `result` - placeholder where to put the result

### quadMult

    private static void quadMult(double[] a,
                                 double[] b,
                                 double[] result)

  Compute (a[0] + a[1]) * (b[0] + b[1]) in extended precision.

  Parameters:
    `a` - first term of the multiplication
    `b` - second term of the multiplication
    `result` - placeholder where to put the result

### expint

    private static double expint(int p,
                                 double[] result)

  Compute exp(p) for a integer p in extended precision.

  Parameters:
    `p` - integer whose exponential is requested
    `result` - placeholder where to put the result in extended precision
  Returns:
    exp(p) in standard precision (equal to result[0] + result[1])

### log

    public static double log(double x)

  Natural logarithm.

  Parameters:
    `x` - a double
  Returns:
    log(x)

### log

    private static double log(double x,
                              double[] hiPrec)

  Internal helper method for natural logarithm function.

  Parameters:
    `x` - original argument of the natural logarithm function
    `hiPrec` - extra bits of precision on output (To Be Confirmed)
  Returns:
    log(x)

### log1p

    public static double log1p(double x)

  Compute log(1 + x).

  Parameters:
    `x` - a number
  Returns:
    log(1 + x)

### log10

    public static double log10(double x)

  Compute the base 10 logarithm.

  Parameters:
    `x` - a number
  Returns:
    log10(x)

### pow

    public static double pow(double x,
                             double y)

  Power function. Compute x^y.

  Parameters:
    `x` - a double
    `y` - a double
  Returns:
    double

### slowLog

    private static double[] slowLog(double xi)

  xi in the range of [1, 2]. 3 5 7 x+1 / x x x \ ln ----- = 2 * | x + ---- + ---- + ---- + ... | 1-x \ 3 5 7 / So, compute a Remez approximation of the following function ln ((sqrt(x)+1)/(1-sqrt(x))) / x This will be an even function with only positive coefficents. x is in the range [0 - 1/3]. Transform xi for input to the above function by setting x = (xi-1)/(xi+1). Input to the polynomial is x^2, then the result is multiplied by x.

  Parameters:
    `xi` - number from which log is requested
  Returns:
    log(xi)

### slowSin

    private static double slowSin(double x,
                                  double[] result)

  For x between 0 and pi/4 compute sine.

  Parameters:
    `x` - number from which sine is requested
    `result` - placeholder where to put the result in extended precision
  Returns:
    sin(x)

### slowCos

    private static double slowCos(double x,
                                  double[] result)

  For x between 0 and pi/4 compute cosine

  Parameters:
    `x` - number from which cosine is requested
    `result` - placeholder where to put the result in extended precision
  Returns:
    cos(x)

### buildSinCosTables

    private static void buildSinCosTables()

  Build the sine and cosine tables.

### polySine

    private static double polySine(double x)

  Computes sin(x) - x, where |x| < 1/16. Use a Remez polynomial approximation.

  Parameters:
    `x` - a number smaller than 1/16
  Returns:
    sin(x) - x

### polyCosine

    private static double polyCosine(double x)

  Computes cos(x) - 1, where |x| < 1/16. Use a Remez polynomial approximation.

  Parameters:
    `x` - a number smaller than 1/16
  Returns:
    cos(x) - 1

### sinQ

    private static double sinQ(double xa,
                               double xb)

  Compute sine over the first quadrant (0 < x < pi/2). Use combination of table lookup and rational polynomial expansion.

  Parameters:
    `xa` - number from which sine is requested
    `xb` - extra bits for x (may be 0.0)
  Returns:
    sin(xa + xb)

### cosQ

    private static double cosQ(double xa,
                               double xb)

  Compute cosine in the first quadrant by subtracting input from PI/2 and then calling sinQ. This is more accurate as the input approaches PI/2.

  Parameters:
    `xa` - number from which cosine is requested
    `xb` - extra bits for x (may be 0.0)
  Returns:
    cos(xa + xb)

### tanQ

    private static double tanQ(double xa,
                               double xb,
                               boolean cotanFlag)

  Compute tangent (or cotangent) over the first quadrant. 0 < x < pi/2 Use combination of table lookup and rational polynomial expansion.

  Parameters:
    `xa` - number from which sine is requested
    `xb` - extra bits for x (may be 0.0)
    `cotanFlag` - if true, compute the cotangent instead of the tangent
  Returns:
    tan(xa+xb) (or cotangent, depending on cotanFlag)

### reducePayneHanek

    private static void reducePayneHanek(double x,
                                         double[] result)

  Reduce the input argument using the Payne and Hanek method. This is good for all inputs 0.0 < x < inf Output is remainder after dividing by PI/2 The result array should contain 3 numbers. result[0] is the integer portion, so mod 4 this gives the quadrant. result[1] is the upper bits of the remainder result[2] is the lower bits of the remainder

  Parameters:
    `x` - number to reduce
    `result` - placeholder where to put the result

### sin

    public static double sin(double x)

  Sine function.

  Parameters:
    `x` - a number
  Returns:
    sin(x)

### cos

    public static double cos(double x)

  Cosine function

  Parameters:
    `x` - a number
  Returns:
    cos(x)

### tan

    public static double tan(double x)

  Tangent function

  Parameters:
    `x` - a number
  Returns:
    tan(x)

### atan

    public static double atan(double x)

  Arctangent function

  Parameters:
    `x` - a number
  Returns:
    atan(x)

### atan

    private static double atan(double xa,
                               double xb,
                               boolean leftPlane)

  Internal helper function to compute arctangent.

  Parameters:
    `xa` - number from which arctangent is requested
    `xb` - extra bits for x (may be 0.0)
    `leftPlane` - if true, result angle must be put in the left half plane
  Returns:
    atan(xa + xb) (or angle shifted by `PI` if leftPlane is true)

### atan2

    public static double atan2(double y,
                               double x)

  Two arguments arctangent function

  Parameters:
    `y` - ordinate
    `x` - abscissa
  Returns:
    phase angle of point (x,y) between `-PI` and `PI`

### asin

    public static double asin(double x)

  Compute the arc sine of a number.

  Parameters:
    `x` - number on which evaluation is done
  Returns:
    arc sine of x

### acos

    public static double acos(double x)

  Compute the arc cosine of a number.

  Parameters:
    `x` - number on which evaluation is done
  Returns:
    arc cosine of x

### cbrt

    public static double cbrt(double x)

  Compute the cubic root of a number.

  Parameters:
    `x` - number on which evaluation is done
  Returns:
    cubic root of x

### toRadians

    public static double toRadians(double x)

  Convert degrees to radians, with error of less than 0.5 ULP

  Parameters:
    `x` - angle in degrees
  Returns:
    x converted into radians

### toDegrees

    public static double toDegrees(double x)

  Convert radians to degrees, with error of less than 0.5 ULP

  Parameters:
    `x` - angle in radians
  Returns:
    x converted into degrees

### abs

    public static int abs(int x)

  Absolute value.

  Parameters:
    `x` - number from which absolute value is requested
  Returns:
    abs(x)

### abs

    public static long abs(long x)

  Absolute value.

  Parameters:
    `x` - number from which absolute value is requested
  Returns:
    abs(x)

### abs

    public static float abs(float x)

  Absolute value.

  Parameters:
    `x` - number from which absolute value is requested
  Returns:
    abs(x)

### abs

    public static double abs(double x)

  Absolute value.

  Parameters:
    `x` - number from which absolute value is requested
  Returns:
    abs(x)

### ulp

    public static double ulp(double x)

  Compute least significant bit (Unit in Last Position) for a number.

  Parameters:
    `x` - number from which ulp is requested
  Returns:
    ulp(x)

### ulp

    public static float ulp(float x)

  Compute least significant bit (Unit in Last Position) for a number.

  Parameters:
    `x` - number from which ulp is requested
  Returns:
    ulp(x)

### scalb

    public static double scalb(double d,
                               int n)

  Multiply a double number by a power of 2.

  Parameters:
    `d` - number to multiply
    `n` - power of 2
  Returns:
    d × 2n

### scalb

    public static float scalb(float f,
                              int n)

  Multiply a float number by a power of 2.

  Parameters:
    `f` - number to multiply
    `n` - power of 2
  Returns:
    f × 2n

### nextAfter

    public static double nextAfter(double d,
                                   double direction)

  Get the next machine representable number after a number, moving in the direction of another number.

  The ordering is as follows (increasing):

  - -INFINITY
  - -MAX_VALUE
  - -MIN_VALUE
  - -0.0
  - +0.0
  - +MIN_VALUE
  - +MAX_VALUE
  - +INFINITY

  If arguments compare equal, then the second argument is returned.

  If `direction` is greater than `d`, the smallest machine representable number strictly greater than `d` is returned; if less, then the largest representable number strictly less than `d` is returned.

  If `d` is infinite and direction does not bring it back to finite numbers, it is returned unchanged.

  Parameters:
    `d` - base number
    `direction` - (the only important thing is whether `direction` is greater or smaller than `d`)
  Returns:
    the next machine representable number in the specified direction

### nextAfter

    public static float nextAfter(float f,
                                  double direction)

  Get the next machine representable number after a number, moving in the direction of another number.

  The ordering is as follows (increasing):

  - -INFINITY
  - -MAX_VALUE
  - -MIN_VALUE
  - -0.0
  - +0.0
  - +MIN_VALUE
  - +MAX_VALUE
  - +INFINITY

  If arguments compare equal, then the second argument is returned.

  If `direction` is greater than `f`, the smallest machine representable number strictly greater than `f` is returned; if less, then the largest representable number strictly less than `f` is returned.

  If `f` is infinite and direction does not bring it back to finite numbers, it is returned unchanged.

  Parameters:
    `f` - base number
    `direction` - (the only important thing is whether `direction` is greater or smaller than `f`)
  Returns:
    the next machine representable number in the specified direction

### floor

    public static double floor(double x)

  Get the largest whole number smaller than x.

  Parameters:
    `x` - number from which floor is requested
  Returns:
    a double number f such that f is an integer f <= x < f + 1.0

### ceil

    public static double ceil(double x)

  Get the smallest whole number larger than x.

  Parameters:
    `x` - number from which ceil is requested
  Returns:
    a double number c such that c is an integer c - 1.0 < x <= c

### rint

    public static double rint(double x)

  Get the whole number that is the nearest to x, or the even one if x is exactly half way between two integers.

  Parameters:
    `x` - number from which nearest whole number is requested
  Returns:
    a double number r such that r is an integer r - 0.5 <= x <= r + 0.5

### round

    public static long round(double x)

  Get the closest long to x.

  Parameters:
    `x` - number from which closest long is requested
  Returns:
    closest long to x

### round

    public static int round(float x)

  Get the closest int to x.

  Parameters:
    `x` - number from which closest int is requested
  Returns:
    closest int to x

### min

    public static int min(int a,
                          int b)

  Compute the minimum of two values

  Parameters:
    `a` - first value
    `b` - second value
  Returns:
    a if a is lesser or equal to b, b otherwise

### min

    public static long min(long a,
                           long b)

  Compute the minimum of two values

  Parameters:
    `a` - first value
    `b` - second value
  Returns:
    a if a is lesser or equal to b, b otherwise

### min

    public static float min(float a,
                            float b)

  Compute the minimum of two values

  Parameters:
    `a` - first value
    `b` - second value
  Returns:
    a if a is lesser or equal to b, b otherwise

### min

    public static double min(double a,
                             double b)

  Compute the minimum of two values

  Parameters:
    `a` - first value
    `b` - second value
  Returns:
    a if a is lesser or equal to b, b otherwise

### max

    public static int max(int a,
                          int b)

  Compute the maximum of two values

  Parameters:
    `a` - first value
    `b` - second value
  Returns:
    b if a is lesser or equal to b, a otherwise

### max

    public static long max(long a,
                           long b)

  Compute the maximum of two values

  Parameters:
    `a` - first value
    `b` - second value
  Returns:
    b if a is lesser or equal to b, a otherwise

### max

    public static float max(float a,
                            float b)

  Compute the maximum of two values

  Parameters:
    `a` - first value
    `b` - second value
  Returns:
    b if a is lesser or equal to b, a otherwise

### max

    public static double max(double a,
                             double b)

  Compute the maximum of two values

  Parameters:
    `a` - first value
    `b` - second value
  Returns:
    b if a is lesser or equal to b, a otherwise

### hypot

    public static double hypot(double x,
                               double y)

  Returns the hypotenuse of a triangle with sides `x` and `y` - sqrt(x2 +y2)
  avoiding intermediate overflow or underflow.
  - If either argument is infinite, then the result is positive infinity.
  - else, if either argument is NaN then the result is NaN.

  Parameters:
    `x` - a value
    `y` - a value
  Returns:
    sqrt(x2 +y2)

### IEEEremainder

    public static double IEEEremainder(double dividend,
                                       double divisor)

  Computes the remainder as prescribed by the IEEE 754 standard. The remainder value is mathematically equal to `x - y*n` where `n` is the mathematical integer closest to the exact mathematical value of the quotient `x/y`. If two mathematical integers are equally close to `x/y` then `n` is the integer that is even.

  - If either operand is NaN, the result is NaN.
  - If the result is not NaN, the sign of the result equals the sign of the dividend.
  - If the dividend is an infinity, or the divisor is a zero, or both, the result is NaN.
  - If the dividend is finite and the divisor is an infinity, the result equals the dividend.
  - If the dividend is a zero and the divisor is finite, the result equals the dividend.

  Note: this implementation currently delegates to `StrictMath.IEEEremainder(double, double)`

  Parameters:
    `dividend` - the number to be divided
    `divisor` - the number by which to divide
  Returns:
    the remainder, rounded

### copySign

    public static double copySign(double magnitude,
                                  double sign)

  Returns the first argument with the sign of the second argument. A NaN `sign` argument is treated as positive.

  Parameters:
    `magnitude` - the value to return
    `sign` - the sign for the returned value
  Returns:
    the magnitude with the same sign as the `sign` argument

### copySign

    public static float copySign(float magnitude,
                                 float sign)

  Returns the first argument with the sign of the second argument. A NaN `sign` argument is treated as positive.

  Parameters:
    `magnitude` - the value to return
    `sign` - the sign for the returned value
  Returns:
    the magnitude with the same sign as the `sign` argument

### getExponent

    public static int getExponent(double d)

  Return the exponent of a double number, removing the bias.

  For double numbers of the form 2x, the unbiased exponent is exactly x.

  Parameters:
    `d` - number from which exponent is requested
  Returns:
    exponent for d in IEEE754 representation, without bias

### getExponent

    public static int getExponent(float f)

  Return the exponent of a float number, removing the bias.

  For float numbers of the form 2x, the unbiased exponent is exactly x.

  Parameters:
    `f` - number from which exponent is requested
  Returns:
    exponent for d in IEEE754 representation, without bias

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