atan.go

     1  // Copyright 2009 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  package math
     6  
     7  /*
     8  	Floating-point arctangent.
     9  */
    10  
    11  // The original C code, the long comment, and the constants below were
    12  // from http://netlib.sandia.gov/cephes/cmath/atan.c, available from
    13  // http://www.netlib.org/cephes/cmath.tgz.
    14  // The go code is a version of the original C.
    15  //
    16  // atan.c
    17  // Inverse circular tangent (arctangent)
    18  //
    19  // SYNOPSIS:
    20  // double x, y, atan();
    21  // y = atan( x );
    22  //
    23  // DESCRIPTION:
    24  // Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
    25  //
    26  // Range reduction is from three intervals into the interval from zero to 0.66.
    27  // The approximant uses a rational function of degree 4/5 of the form
    28  // x + x**3 P(x)/Q(x).
    29  //
    30  // ACCURACY:
    31  //                      Relative error:
    32  // arithmetic   domain    # trials  peak     rms
    33  //    DEC       -10, 10   50000     2.4e-17  8.3e-18
    34  //    IEEE      -10, 10   10^6      1.8e-16  5.0e-17
    35  //
    36  // Cephes Math Library Release 2.8:  June, 2000
    37  // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
    38  //
    39  // The readme file at http://netlib.sandia.gov/cephes/ says:
    40  //    Some software in this archive may be from the book _Methods and
    41  // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
    42  // International, 1989) or from the Cephes Mathematical Library, a
    43  // commercial product. In either event, it is copyrighted by the author.
    44  // What you see here may be used freely but it comes with no support or
    45  // guarantee.
    46  //
    47  //   The two known misprints in the book are repaired here in the
    48  // source listings for the gamma function and the incomplete beta
    49  // integral.
    50  //
    51  //   Stephen L. Moshier
    52  //   moshier@na-net.ornl.gov
    53  
    54  // xatan evaluates a series valid in the range [0, 0.66].
    55  func xatan(x float64) float64 {
    56  	const (
    57  		P0 = -8.750608600031904122785e-01
    58  		P1 = -1.615753718733365076637e+01
    59  		P2 = -7.500855792314704667340e+01
    60  		P3 = -1.228866684490136173410e+02
    61  		P4 = -6.485021904942025371773e+01
    62  		Q0 = +2.485846490142306297962e+01
    63  		Q1 = +1.650270098316988542046e+02
    64  		Q2 = +4.328810604912902668951e+02
    65  		Q3 = +4.853903996359136964868e+02
    66  		Q4 = +1.945506571482613964425e+02
    67  	)
    68  	z := x * x
    69  	z = z * ((((P0*z+P1)*z+P2)*z+P3)*z + P4) / (((((z+Q0)*z+Q1)*z+Q2)*z+Q3)*z + Q4)
    70  	z = x*z + x
    71  	return z
    72  }
    73  
    74  // satan reduces its argument (known to be positive)
    75  // to the range [0, 0.66] and calls xatan.
    76  func satan(x float64) float64 {
    77  	const (
    78  		Morebits = 6.123233995736765886130e-17 // pi/2 = PIO2 + Morebits
    79  		Tan3pio8 = 2.41421356237309504880      // tan(3*pi/8)
    80  	)
    81  	if x <= 0.66 {
    82  		return xatan(x)
    83  	}
    84  	if x > Tan3pio8 {
    85  		return Pi/2 - xatan(1/x) + Morebits
    86  	}
    87  	return Pi/4 + xatan((x-1)/(x+1)) + 0.5*Morebits
    88  }
    89  
    90  // Atan returns the arctangent, in radians, of x.
    91  //
    92  // Special cases are:
    93  //
    94  //	Atan(±0) = ±0
    95  //	Atan(±Inf) = ±Pi/2
    96  func Atan(x float64) float64 {
    97  	if haveArchAtan {
    98  		return archAtan(x)
    99  	}
   100  	return atan(x)
   101  }
   102  
   103  func atan(x float64) float64 {
   104  	if x == 0 {
   105  		return x
   106  	}
   107  	if x > 0 {
   108  		return satan(x)
   109  	}
   110  	return -satan(-x)
   111  }
   112
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