/*
Floating-point arctangent.
-
- Atan returns the value of the arctangent of its
- argument in the range [-pi/2,pi/2].
- There are no error returns.
- Coefficients are #5077 from Hart & Cheney. (19.56D)
*/
-// xatan evaluates a series valid in the
-// range [-0.414...,+0.414...]. (tan(pi/8))
-func xatan(arg float64) float64 {
+// The original C code, the long comment, and the constants below were
+// from http://netlib.sandia.gov/cephes/cmath/atan.c, available from
+// http://www.netlib.org/cephes/cmath.tgz.
+// The go code is a version of the original C.
+//
+// atan.c
+// Inverse circular tangent (arctangent)
+//
+// SYNOPSIS:
+// double x, y, atan();
+// y = atan( x );
+//
+// DESCRIPTION:
+// Returns radian angle between -pi/2 and +pi/2 whose tangent is x.
+//
+// Range reduction is from three intervals into the interval from zero to 0.66.
+// The approximant uses a rational function of degree 4/5 of the form
+// x + x**3 P(x)/Q(x).
+//
+// ACCURACY:
+// Relative error:
+// arithmetic domain # trials peak rms
+// DEC -10, 10 50000 2.4e-17 8.3e-18
+// IEEE -10, 10 10^6 1.8e-16 5.0e-17
+//
+// Cephes Math Library Release 2.8: June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+// Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+// The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+// Stephen L. Moshier
+// moshier@na-net.ornl.gov
+
+// xatan evaluates a series valid in the range [0, 0.66].
+func xatan(x float64) float64 {
const (
- P4 = .161536412982230228262e2
- P3 = .26842548195503973794141e3
- P2 = .11530293515404850115428136e4
- P1 = .178040631643319697105464587e4
- P0 = .89678597403663861959987488e3
- Q4 = .5895697050844462222791e2
- Q3 = .536265374031215315104235e3
- Q2 = .16667838148816337184521798e4
- Q1 = .207933497444540981287275926e4
- Q0 = .89678597403663861962481162e3
+ P0 = -8.750608600031904122785e-01
+ P1 = -1.615753718733365076637e+01
+ P2 = -7.500855792314704667340e+01
+ P3 = -1.228866684490136173410e+02
+ P4 = -6.485021904942025371773e+01
+ Q0 = +2.485846490142306297962e+01
+ Q1 = +1.650270098316988542046e+02
+ Q2 = +4.328810604912902668951e+02
+ Q3 = +4.853903996359136964868e+02
+ Q4 = +1.945506571482613964425e+02
)
- sq := arg * arg
- value := ((((P4*sq+P3)*sq+P2)*sq+P1)*sq + P0)
- value = value / (((((sq+Q4)*sq+Q3)*sq+Q2)*sq+Q1)*sq + Q0)
- return value * arg
+ z := x * x
+ z = z * ((((P0*z+P1)*z+P2)*z+P3)*z + P4) / (((((z+Q0)*z+Q1)*z+Q2)*z+Q3)*z + Q4)
+ z = x*z + x
+ return z
}
// satan reduces its argument (known to be positive)
-// to the range [0,0.414...] and calls xatan.
-func satan(arg float64) float64 {
- if arg < Sqrt2-1 {
- return xatan(arg)
+// to the range [0, 0.66] and calls xatan.
+func satan(x float64) float64 {
+ const (
+ Morebits = 6.123233995736765886130e-17 // pi/2 = PIO2 + Morebits
+ Tan3pio8 = 2.41421356237309504880 // tan(3*pi/8)
+ )
+ if x <= 0.66 {
+ return xatan(x)
}
- if arg > Sqrt2+1 {
- return Pi/2 - xatan(1/arg)
+ if x > Tan3pio8 {
+ return Pi/2 - xatan(1/x) + Morebits
}
- return Pi/4 + xatan((arg-1)/(arg+1))
+ return Pi/4 + xatan((x-1)/(x+1)) + 0.5*Morebits
}
// Atan returns the arctangent of x.
//
// Special cases are:
-// Atan(±0) = ±0
-// Atan(±Inf) = ±Pi/2
+// Atan(±0) = ±0
+// Atan(±Inf) = ±Pi/2
func Atan(x float64) float64
func atan(x float64) float64 {