// Copyright 2009 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package math
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/*
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Floating-point arctangent.
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*/
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// The original C code, the long comment, and the constants below were
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// from http://netlib.sandia.gov/cephes/cmath/atan.c, available from
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// http://www.netlib.org/cephes/cmath.tgz.
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// The go code is a version of the original C.
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//
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// atan.c
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// Inverse circular tangent (arctangent)
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//
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// SYNOPSIS:
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// double x, y, atan();
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// y = atan( x );
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//
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// DESCRIPTION:
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// Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
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//
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// Range reduction is from three intervals into the interval from zero to 0.66.
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// The approximant uses a rational function of degree 4/5 of the form
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// x + x**3 P(x)/Q(x).
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//
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// ACCURACY:
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// Relative error:
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// arithmetic domain # trials peak rms
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// DEC -10, 10 50000 2.4e-17 8.3e-18
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// IEEE -10, 10 10^6 1.8e-16 5.0e-17
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//
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// Cephes Math Library Release 2.8: June, 2000
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// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
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//
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// The readme file at http://netlib.sandia.gov/cephes/ says:
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// Some software in this archive may be from the book _Methods and
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// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
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// International, 1989) or from the Cephes Mathematical Library, a
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// commercial product. In either event, it is copyrighted by the author.
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// What you see here may be used freely but it comes with no support or
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// guarantee.
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//
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// The two known misprints in the book are repaired here in the
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// source listings for the gamma function and the incomplete beta
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// integral.
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//
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// Stephen L. Moshier
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// moshier@na-net.ornl.gov
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// xatan evaluates a series valid in the range [0, 0.66].
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func xatan(x float64) float64 {
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const (
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P0 = -8.750608600031904122785e-01
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P1 = -1.615753718733365076637e+01
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P2 = -7.500855792314704667340e+01
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P3 = -1.228866684490136173410e+02
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P4 = -6.485021904942025371773e+01
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Q0 = +2.485846490142306297962e+01
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Q1 = +1.650270098316988542046e+02
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Q2 = +4.328810604912902668951e+02
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Q3 = +4.853903996359136964868e+02
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Q4 = +1.945506571482613964425e+02
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)
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z := x * x
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z = z * ((((P0*z+P1)*z+P2)*z+P3)*z + P4) / (((((z+Q0)*z+Q1)*z+Q2)*z+Q3)*z + Q4)
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z = x*z + x
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return z
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}
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// satan reduces its argument (known to be positive)
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// to the range [0, 0.66] and calls xatan.
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func satan(x float64) float64 {
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const (
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Morebits = 6.123233995736765886130e-17 // pi/2 = PIO2 + Morebits
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Tan3pio8 = 2.41421356237309504880 // tan(3*pi/8)
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)
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if x <= 0.66 {
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return xatan(x)
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}
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if x > Tan3pio8 {
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return Pi/2 - xatan(1/x) + Morebits
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}
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return Pi/4 + xatan((x-1)/(x+1)) + 0.5*Morebits
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}
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// Atan returns the arctangent, in radians, of x.
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//
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// Special cases are:
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// Atan(±0) = ±0
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// Atan(±Inf) = ±Pi/2
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func Atan(x float64) float64
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func atan(x float64) float64 {
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if x == 0 {
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return x
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}
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if x > 0 {
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return satan(x)
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}
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return -satan(-x)
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}
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