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
5package 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//   [email protected]
53
54// xatan evaluates a series valid in the range [0, 0.66].
55func 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.
76func 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
96func Atan(x float64) float64 {
97	if haveArchAtan {
98		return archAtan(x)
99	}
100	return atan(x)
101}
102
103func 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