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