module math // 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/ctgz. // 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/2.0 and +pi/2.0 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 // pi/2.0 = PIO2 + morebits // tan3pio8 = tan(3*pi/8) const ( morebits = 6.123233995736765886130e-17 tan3pio8 = 2.41421356237309504880 ) // xatan evaluates a series valid in the range [0, 0.66]. [inline] fn xatan(x f64) f64 { xatan_p0 := -8.750608600031904122785e-01 xatan_p1 := -1.615753718733365076637e+01 xatan_p2 := -7.500855792314704667340e+01 xatan_p3 := -1.228866684490136173410e+02 xatan_p4 := -6.485021904942025371773e+01 xatan_q0 := 2.485846490142306297962e+01 xatan_q1 := 1.650270098316988542046e+02 xatan_q2 := 4.328810604912902668951e+02 xatan_q3 := 4.853903996359136964868e+02 xatan_q4 := 1.945506571482613964425e+02 mut z := x * x z = z * ((((xatan_p0 * z + xatan_p1) * z + xatan_p2) * z + xatan_p3) * z + xatan_p4) / (((((z + xatan_q0) * z + xatan_q1) * z + xatan_q2) * z + xatan_q3) * z + xatan_q4) z = x * z + x return z } // satan reduces its argument (known to be positive) // to the range [0, 0.66] and calls xatan. [inline] fn satan(x f64) f64 { if x <= 0.66 { return xatan(x) } if x > math.tan3pio8 { return pi / 2.0 - xatan(1.0 / x) + f64(math.morebits) } return pi / 4 + xatan((x - 1.0) / (x + 1.0)) + 0.5 * f64(math.morebits) } // atan returns the arctangent, in radians, of x. // // special cases are: // atan(±0) = ±0 // atan(±inf) = ±pi/2.0 pub fn atan(x f64) f64 { if x == 0 { return x } if x > 0 { return satan(x) } return -satan(-x) } // atan2 returns the arc tangent of y/x, using // the signs of the two to determine the quadrant // of the return value. // // special cases are (in order): // atan2(y, nan) = nan // atan2(nan, x) = nan // atan2(+0, x>=0) = +0 // atan2(-0, x>=0) = -0 // atan2(+0, x<=-0) = +pi // atan2(-0, x<=-0) = -pi // atan2(y>0, 0) = +pi/2.0 // atan2(y<0, 0) = -pi/2.0 // atan2(+inf, +inf) = +pi/4 // atan2(-inf, +inf) = -pi/4 // atan2(+inf, -inf) = 3pi/4 // atan2(-inf, -inf) = -3pi/4 // atan2(y, +inf) = 0 // atan2(y>0, -inf) = +pi // atan2(y<0, -inf) = -pi // atan2(+inf, x) = +pi/2.0 // atan2(-inf, x) = -pi/2.0 pub fn atan2(y f64, x f64) f64 { // special cases if is_nan(y) || is_nan(x) { return nan() } if y == 0.0 { if x >= 0 && !signbit(x) { return copysign(0, y) } return copysign(pi, y) } if x == 0.0 { return copysign(pi / 2.0, y) } if is_inf(x, 0) { if is_inf(x, 1) { if is_inf(y, 0) { return copysign(pi / 4, y) } return copysign(0, y) } if is_inf(y, 0) { return copysign(3.0 * pi / 4.0, y) } return copysign(pi, y) } if is_inf(y, 0) { return copysign(pi / 2.0, y) } // Call atan and determine the quadrant. q := atan(y / x) if x < 0 { if q <= 0 { return q + pi } return q - pi } return q } /* Floating-point arcsine and arccosine. They are implemented by computing the arctangent after appropriate range reduction. */ // asin returns the arcsine, in radians, of x. // // special cases are: // asin(±0) = ±0 // asin(x) = nan if x < -1 or x > 1 pub fn asin(x_ f64) f64 { mut x := x_ if x == 0.0 { return x // special case } mut sign := false if x < 0.0 { x = -x sign = true } if x > 1.0 { return nan() // special case } mut temp := sqrt(1.0 - x * x) if x > 0.7 { temp = pi / 2.0 - satan(temp / x) } else { temp = satan(x / temp) } if sign { temp = -temp } return temp } // acos returns the arccosine, in radians, of x. // // special case is: // acos(x) = nan if x < -1 or x > 1 [inline] pub fn acos(x f64) f64 { if (x < -1.0) || (x > 1.0) { return nan() } if x == 0.0 { return nan() } if x > 0.5 { return f64(2.0) * asin(sqrt(0.5 - 0.5 * x)) } mut z := pi / f64(4.0) - asin(x) z = z + math.morebits z = z + pi / f64(4.0) return z }