sincos.go

// Copyright 2011 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package math32

import (
	"math/bits"
)

/*
	Floating-point sine and cosine.
*/

// The original C code, the long comment, and the constants
// below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
// available from http://www.netlib.org/cephes/cmath.tgz.
// The go code is a simplified version of the original C.
//
//      sin.c
//
//      Circular sine
//
// SYNOPSIS:
//
// double x, y, sin();
// y = sin( x );
//
// DESCRIPTION:
//
// Range reduction is into intervals of pi/4.  The reduction error is nearly
// eliminated by contriving an extended precision modular arithmetic.
//
// Two polynomial approximating functions are employed.
// Between 0 and pi/4 the sine is approximated by
//      x  +  x**3 P(x**2).
// Between pi/4 and pi/2 the cosine is represented as
//      1  -  x**2 Q(x**2).
//
// ACCURACY:
//
//                      Relative error:
// arithmetic   domain      # trials      peak         rms
//    DEC       0, 10       150000       3.0e-17     7.8e-18
//    IEEE -1.07e9,+1.07e9  130000       2.1e-16     5.4e-17
//
// Partial loss of accuracy begins to occur at x = 2**30 = 1.074e9.  The loss
// is not gradual, but jumps suddenly to about 1 part in 10e7.  Results may
// be meaningless for x > 2**49 = 5.6e14.
//
//      cos.c
//
//      Circular cosine
//
// SYNOPSIS:
//
// double x, y, cos();
// y = cos( x );
//
// DESCRIPTION:
//
// Range reduction is into intervals of pi/4.  The reduction error is nearly
// eliminated by contriving an extended precision modular arithmetic.
//
// Two polynomial approximating functions are employed.
// Between 0 and pi/4 the cosine is approximated by
//      1  -  x**2 Q(x**2).
// Between pi/4 and pi/2 the sine is represented as
//      x  +  x**3 P(x**2).
//
// ACCURACY:
//
//                      Relative error:
// arithmetic   domain      # trials      peak         rms
//    IEEE -1.07e9,+1.07e9  130000       2.1e-16     5.4e-17
//    DEC        0,+1.07e9   17000       3.0e-17     7.2e-18
//
// 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

// sin coefficients
var _sin = [...]float32{
	1.58962301576546568060e-10, // 0x3de5d8fd1fd19ccd
	-2.50507477628578072866e-8, // 0xbe5ae5e5a9291f5d
	2.75573136213857245213e-6,  // 0x3ec71de3567d48a1
	-1.98412698295895385996e-4, // 0xbf2a01a019bfdf03
	8.33333333332211858878e-3,  // 0x3f8111111110f7d0
	-1.66666666666666307295e-1, // 0xbfc5555555555548
}

// cos coefficients
var _cos = [...]float32{
	-1.13585365213876817300e-11, // 0xbda8fa49a0861a9b
	2.08757008419747316778e-9,   // 0x3e21ee9d7b4e3f05
	-2.75573141792967388112e-7,  // 0xbe927e4f7eac4bc6
	2.48015872888517045348e-5,   // 0x3efa01a019c844f5
	-1.38888888888730564116e-3,  // 0xbf56c16c16c14f91
	4.16666666666665929218e-2,   // 0x3fa555555555554b
}

// Sincos returns Sin(x), Cos(x).
//
// Special cases are:
//
//	Sincos(±0) = ±0, 1
//	Sincos(±Inf) = NaN, NaN
//	Sincos(NaN) = NaN, NaN
func Sincos(x float32) (sin, cos float32) {
	const (
		PI4A = 7.85398125648498535156e-1  // 0x3fe921fb40000000, Pi/4 split into three parts
		PI4B = 3.77489470793079817668e-8  // 0x3e64442d00000000,
		PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170,
	)
	// special cases
	switch {
	case x == 0:
		return x, 1 // return ±0.0, 1.0
	case IsNaN(x) || IsInf(x, 0):
		return NaN(), NaN()
	}

	// make argument positive
	sinSign, cosSign := false, false
	if x < 0 {
		x = -x
		sinSign = true
	}

	var j uint64
	var y, z float32
	if x >= reduceThreshold {
		j, z = trigReduce(x)
	} else {
		j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle
		y = float32(j)           // integer part of x/(Pi/4), as float

		if j&1 == 1 { // map zeros to origin
			j++
			y++
		}
		j &= 7                               // octant modulo 2Pi radians (360 degrees)
		z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
	}
	if j > 3 { // reflect in x axis
		j -= 4
		sinSign, cosSign = !sinSign, !cosSign
	}
	if j > 1 {
		cosSign = !cosSign
	}

	zz := z * z
	cos = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
	sin = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
	if j == 1 || j == 2 {
		sin, cos = cos, sin
	}
	if cosSign {
		cos = -cos
	}
	if sinSign {
		sin = -sin
	}
	return
}

// Sin returns the sine of the radian argument x.
//
// Special cases are:
//
//	Sin(±0) = ±0
//	Sin(±Inf) = NaN
//	Sin(NaN) = NaN
func Sin(x float32) float32 {
	const (
		PI4A = 7.85398125648498535156e-1  // 0x3fe921fb40000000, Pi/4 split into three parts
		PI4B = 3.77489470793079817668e-8  // 0x3e64442d00000000,
		PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170,
	)
	// special cases
	switch {
	case x == 0 || IsNaN(x):
		return x // return ±0 || NaN()
	case IsInf(x, 0):
		return NaN()
	}

	// make argument positive but save the sign
	sign := false
	if x < 0 {
		x = -x
		sign = true
	}

	var j uint64
	var y, z float32
	if x >= reduceThreshold {
		j, z = trigReduce(x)
	} else {
		j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle
		y = float32(j)           // integer part of x/(Pi/4), as float

		// map zeros to origin
		if j&1 == 1 {
			j++
			y++
		}
		j &= 7                               // octant modulo 2Pi radians (360 degrees)
		z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
	}
	// reflect in x axis
	if j > 3 {
		sign = !sign
		j -= 4
	}
	zz := z * z
	if j == 1 || j == 2 {
		y = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
	} else {
		y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
	}
	if sign {
		y = -y
	}
	return y
}

// Cos returns the cosine of the radian argument x.
//
// Special cases are:
//
//	Cos(±Inf) = NaN
//	Cos(NaN) = NaN
func Cos(x float32) float32 {
	const (
		PI4A = 7.85398125648498535156e-1  // 0x3fe921fb40000000, Pi/4 split into three parts
		PI4B = 3.77489470793079817668e-8  // 0x3e64442d00000000,
		PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170,
	)
	// special cases
	switch {
	case IsNaN(x) || IsInf(x, 0):
		return NaN()
	}

	// make argument positive
	sign := false
	x = Abs(x)

	var j uint64
	var y, z float32
	if x >= reduceThreshold {
		j, z = trigReduce(x)
	} else {
		j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle
		y = float32(j)           // integer part of x/(Pi/4), as float

		// map zeros to origin
		if j&1 == 1 {
			j++
			y++
		}
		j &= 7                               // octant modulo 2Pi radians (360 degrees)
		z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
	}

	if j > 3 {
		j -= 4
		sign = !sign
	}
	if j > 1 {
		sign = !sign
	}

	zz := z * z
	if j == 1 || j == 2 {
		y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
	} else {
		y = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
	}
	if sign {
		y = -y
	}
	return y
}

// reduceThreshold is the maximum value of x where the reduction using Pi/4
// in 3 float64 parts still gives accurate results. This threshold
// is set by y*C being representable as a float64 without error
// where y is given by y = floor(x * (4 / Pi)) and C is the leading partial
// terms of 4/Pi. Since the leading terms (PI4A and PI4B in sin.go) have 30
// and 32 trailing zero bits, y should have less than 30 significant bits.
//
//	y < 1<<30  -> floor(x*4/Pi) < 1<<30 -> x < (1<<30 - 1) * Pi/4
//
// So, conservatively we can take x < 1<<29.
// Above this threshold Payne-Hanek range reduction must be used.
const reduceThreshold = 1 << 29

// trigReduce implements Payne-Hanek range reduction by Pi/4
// for x > 0. It returns the integer part mod 8 (j) and
// the fractional part (z) of x / (Pi/4).
// The implementation is based on:
// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
// K. C. Ng et al, March 24, 1992
// The simulated multi-precision calculation of x*B uses 64-bit integer arithmetic.
func trigReduce(x float32) (j uint64, z float32) {
	const PI4 = Pi / 4
	if x < PI4 {
		return 0, x
	}
	// Extract out the integer and exponent such that,
	// x = ix * 2 ** exp.
	ix := Float32bits(x)
	exp := int(ix>>shift&mask) - bias - shift
	ix &^= mask << shift
	ix |= 1 << shift
	// Use the exponent to extract the 3 appropriate uint64 digits from mPi4,
	// B ~ (z0, z1, z2), such that the product leading digit has the exponent -61.
	// Note, exp >= -53 since x >= PI4 and exp < 971 for maximum float64.
	const floatingbits = 32 - 3
	digit, bitshift := uint(exp+floatingbits)/32, uint(exp+floatingbits)%32
	z0 := (mPi4[digit] << bitshift) | (mPi4[digit+1] >> (32 - bitshift))
	z1 := (mPi4[digit+1] << bitshift) | (mPi4[digit+2] >> (32 - bitshift))
	z2 := (mPi4[digit+2] << bitshift) | (mPi4[digit+3] >> (32 - bitshift))
	// Multiply mantissa by the digits and extract the upper two digits (hi, lo).
	z2hi, _ := bits.Mul64(z2, uint64(ix))
	z1hi, z1lo := bits.Mul64(z1, uint64(ix))
	z0lo := z0 * uint64(ix)
	lo, c := bits.Add64(z1lo, z2hi, 0)
	hi, _ := bits.Add64(z0lo, z1hi, c)
	// The top 3 bits are j.
	j = hi >> floatingbits
	// Extract the fraction and find its magnitude.
	hi = hi<<3 | lo>>floatingbits
	lz := uint(bits.LeadingZeros64(hi))
	e := uint64(bias - (lz + 1))
	// Clear implicit mantissa bit and shift into place.
	hi = (hi << (lz + 1)) | (lo >> (32 - (lz + 1)))
	hi >>= 43 - shift
	// Include the exponent and convert to a float.
	hi |= e << shift
	z = Float32frombits(uint32(hi))
	// Map zeros to origin.
	if j&1 == 1 {
		j++
		j &= 7
		z--
	}
	// Multiply the fractional part by pi/4.
	return j, z * PI4
}

// mPi4 is the binary digits of 4/pi as a uint64 array,
// that is, 4/pi = Sum mPi4[i]*2^(-64*i)
// 19 64-bit digits and the leading one bit give 1217 bits
// of precision to handle the largest possible float64 exponent.
var mPi4 = [...]uint64{
	0x0000000000000001,
	0x45f306dc9c882a53,
	0xf84eafa3ea69bb81,
	0xb6c52b3278872083,
	0xfca2c757bd778ac3,
	0x6e48dc74849ba5c0,
	0x0c925dd413a32439,
	0xfc3bd63962534e7d,
	0xd1046bea5d768909,
	0xd338e04d68befc82,
	0x7323ac7306a673e9,
	0x3908bf177bf25076,
	0x3ff12fffbc0b301f,
	0xde5e2316b414da3e,
	0xda6cfd9e4f96136e,
	0x9e8c7ecd3cbfd45a,
	0xea4f758fd7cbe2f6,
	0x7a0e73ef14a525d4,
	0xd7f6bf623f1aba10,
	0xac06608df8f6d757,
}