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 // Atan(±0) = ±0 94 // Atan(±Inf) = ±Pi/2 95 func Atan(x float64) float64 96 97 func atan(x float64) float64 { 98 if x == 0 { 99 return x 100 } 101 if x > 0 { 102 return satan(x) 103 } 104 return -satan(-x) 105 } 106