Source file src/pkg/math/sin.go

     1	// Copyright 2011 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	
     5	package math
     6	
     7	/*
     8		Floating-point sine and cosine.
     9	*/
    10	
    11	// The original C code, the long comment, and the constants
    12	// below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
    13	// available from http://www.netlib.org/cephes/cmath.tgz.
    14	// The go code is a simplified version of the original C.
    15	//
    16	//      sin.c
    17	//
    18	//      Circular sine
    19	//
    20	// SYNOPSIS:
    21	//
    22	// double x, y, sin();
    23	// y = sin( x );
    24	//
    25	// DESCRIPTION:
    26	//
    27	// Range reduction is into intervals of pi/4.  The reduction error is nearly
    28	// eliminated by contriving an extended precision modular arithmetic.
    29	//
    30	// Two polynomial approximating functions are employed.
    31	// Between 0 and pi/4 the sine is approximated by
    32	//      x  +  x**3 P(x**2).
    33	// Between pi/4 and pi/2 the cosine is represented as
    34	//      1  -  x**2 Q(x**2).
    35	//
    36	// ACCURACY:
    37	//
    38	//                      Relative error:
    39	// arithmetic   domain      # trials      peak         rms
    40	//    DEC       0, 10       150000       3.0e-17     7.8e-18
    41	//    IEEE -1.07e9,+1.07e9  130000       2.1e-16     5.4e-17
    42	//
    43	// Partial loss of accuracy begins to occur at x = 2**30 = 1.074e9.  The loss
    44	// is not gradual, but jumps suddenly to about 1 part in 10e7.  Results may
    45	// be meaningless for x > 2**49 = 5.6e14.
    46	//
    47	//      cos.c
    48	//
    49	//      Circular cosine
    50	//
    51	// SYNOPSIS:
    52	//
    53	// double x, y, cos();
    54	// y = cos( x );
    55	//
    56	// DESCRIPTION:
    57	//
    58	// Range reduction is into intervals of pi/4.  The reduction error is nearly
    59	// eliminated by contriving an extended precision modular arithmetic.
    60	//
    61	// Two polynomial approximating functions are employed.
    62	// Between 0 and pi/4 the cosine is approximated by
    63	//      1  -  x**2 Q(x**2).
    64	// Between pi/4 and pi/2 the sine is represented as
    65	//      x  +  x**3 P(x**2).
    66	//
    67	// ACCURACY:
    68	//
    69	//                      Relative error:
    70	// arithmetic   domain      # trials      peak         rms
    71	//    IEEE -1.07e9,+1.07e9  130000       2.1e-16     5.4e-17
    72	//    DEC        0,+1.07e9   17000       3.0e-17     7.2e-18
    73	//
    74	// Cephes Math Library Release 2.8:  June, 2000
    75	// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
    76	//
    77	// The readme file at http://netlib.sandia.gov/cephes/ says:
    78	//    Some software in this archive may be from the book _Methods and
    79	// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
    80	// International, 1989) or from the Cephes Mathematical Library, a
    81	// commercial product. In either event, it is copyrighted by the author.
    82	// What you see here may be used freely but it comes with no support or
    83	// guarantee.
    84	//
    85	//   The two known misprints in the book are repaired here in the
    86	// source listings for the gamma function and the incomplete beta
    87	// integral.
    88	//
    89	//   Stephen L. Moshier
    90	//   [email protected]
    91	
    92	// sin coefficients
    93	var _sin = [...]float64{
    94		1.58962301576546568060E-10, // 0x3de5d8fd1fd19ccd
    95		-2.50507477628578072866E-8, // 0xbe5ae5e5a9291f5d
    96		2.75573136213857245213E-6,  // 0x3ec71de3567d48a1
    97		-1.98412698295895385996E-4, // 0xbf2a01a019bfdf03
    98		8.33333333332211858878E-3,  // 0x3f8111111110f7d0
    99		-1.66666666666666307295E-1, // 0xbfc5555555555548
   100	}
   101	
   102	// cos coefficients
   103	var _cos = [...]float64{
   104		-1.13585365213876817300E-11, // 0xbda8fa49a0861a9b
   105		2.08757008419747316778E-9,   // 0x3e21ee9d7b4e3f05
   106		-2.75573141792967388112E-7,  // 0xbe927e4f7eac4bc6
   107		2.48015872888517045348E-5,   // 0x3efa01a019c844f5
   108		-1.38888888888730564116E-3,  // 0xbf56c16c16c14f91
   109		4.16666666666665929218E-2,   // 0x3fa555555555554b
   110	}
   111	
   112	// Cos returns the cosine of x.
   113	//
   114	// Special cases are:
   115	//	Cos(±Inf) = NaN
   116	//	Cos(NaN) = NaN
   117	func Cos(x float64) float64
   118	
   119	func cos(x float64) float64 {
   120		const (
   121			PI4A = 7.85398125648498535156E-1                             // 0x3fe921fb40000000, Pi/4 split into three parts
   122			PI4B = 3.77489470793079817668E-8                             // 0x3e64442d00000000,
   123			PI4C = 2.69515142907905952645E-15                            // 0x3ce8469898cc5170,
   124			M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi
   125		)
   126		// special cases
   127		switch {
   128		case IsNaN(x) || IsInf(x, 0):
   129			return NaN()
   130		}
   131	
   132		// make argument positive
   133		sign := false
   134		if x < 0 {
   135			x = -x
   136		}
   137	
   138		j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle
   139		y := float64(j)      // integer part of x/(Pi/4), as float
   140	
   141		// map zeros to origin
   142		if j&1 == 1 {
   143			j += 1
   144			y += 1
   145		}
   146		j &= 7 // octant modulo 2Pi radians (360 degrees)
   147		if j > 3 {
   148			j -= 4
   149			sign = !sign
   150		}
   151		if j > 1 {
   152			sign = !sign
   153		}
   154	
   155		z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
   156		zz := z * z
   157		if j == 1 || j == 2 {
   158			y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
   159		} else {
   160			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])
   161		}
   162		if sign {
   163			y = -y
   164		}
   165		return y
   166	}
   167	
   168	// Sin returns the sine of x.
   169	//
   170	// Special cases are:
   171	//	Sin(±0) = ±0
   172	//	Sin(±Inf) = NaN
   173	//	Sin(NaN) = NaN
   174	func Sin(x float64) float64
   175	
   176	func sin(x float64) float64 {
   177		const (
   178			PI4A = 7.85398125648498535156E-1                             // 0x3fe921fb40000000, Pi/4 split into three parts
   179			PI4B = 3.77489470793079817668E-8                             // 0x3e64442d00000000,
   180			PI4C = 2.69515142907905952645E-15                            // 0x3ce8469898cc5170,
   181			M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi
   182		)
   183		// special cases
   184		switch {
   185		case x == 0 || IsNaN(x):
   186			return x // return ±0 || NaN()
   187		case IsInf(x, 0):
   188			return NaN()
   189		}
   190	
   191		// make argument positive but save the sign
   192		sign := false
   193		if x < 0 {
   194			x = -x
   195			sign = true
   196		}
   197	
   198		j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle
   199		y := float64(j)      // integer part of x/(Pi/4), as float
   200	
   201		// map zeros to origin
   202		if j&1 == 1 {
   203			j += 1
   204			y += 1
   205		}
   206		j &= 7 // octant modulo 2Pi radians (360 degrees)
   207		// reflect in x axis
   208		if j > 3 {
   209			sign = !sign
   210			j -= 4
   211		}
   212	
   213		z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
   214		zz := z * z
   215		if j == 1 || j == 2 {
   216			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])
   217		} else {
   218			y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
   219		}
   220		if sign {
   221			y = -y
   222		}
   223		return y
   224	}