Algorithm to find the sin function: Input the angle in degrees. Convert the angle from degrees to radians using the formula radians = degrees * pi / 180. Set a variable sum to 0. Set a variable term to the angle in radians. Set a variable sign to 1. For i = 1 to n (where n is the number of terms to be used): a. Add the term to the sum using the formula sum = sum + sign * term. b. Multiply the term by -1 * angle^2 / (2 * i * (2 * i - 1)). c. Change the sign of the term to its opposite.
Output the value of the sum.
/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
* Copyright (C) 2001-2024 Free Software Foundation, Inc.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as published by
* the Free Software Foundation; either version 2.1 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with this program; if not, see <https://www.gnu.org/licenses/>.
*/
/****************************************************************************/
/* */
/* MODULE_NAME:usncs.c */
/* */
/* FUNCTIONS: usin */
/* ucos */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h usncs.h */
/* branred.c sincos.tbl */
/* */
/* An ultimate sin and cos routine. Given an IEEE double machine number x */
/* it computes sin(x) or cos(x) with ~0.55 ULP. */
/* Assumption: Machine arithmetic operations are performed in */
/* round to nearest mode of IEEE 754 standard. */
/* */
/****************************************************************************/
#include <errno.h>
#include <float.h>
#include "endian.h"
#include "mydefs.h"
#include "usncs.h"
#include <math.h>
#include <math_private.h>
#include <fenv_private.h>
#include <math-underflow.h>
#include <libm-alias-double.h>
#include <fenv.h>
/* Helper macros to compute sin of the input values. */
#define POLYNOMIAL2(xx) ((((s5 * (xx) + s4) * (xx) + s3) * (xx) + s2) * (xx))
#define POLYNOMIAL(xx) (POLYNOMIAL2 (xx) + s1)
/* The computed polynomial is a variation of the Taylor series expansion for
sin(x):
x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - dx*x^2/2 + dx
The constants s1, s2, s3, etc. are pre-computed values of 1/3!, 1/5! and so
on. The result is returned to LHS. */
#define TAYLOR_SIN(xx, x, dx) \
({ \
double t = ((POLYNOMIAL (xx) * (x) - 0.5 * (dx)) * (xx) + (dx)); \
double res = (x) + t; \
res; \
})
#define SINCOS_TABLE_LOOKUP(u, sn, ssn, cs, ccs) \
({ \
int4 k = u.i[LOW_HALF] << 2; \
sn = __sincostab.x[k]; \
ssn = __sincostab.x[k + 1]; \
cs = __sincostab.x[k + 2]; \
ccs = __sincostab.x[k + 3]; \
})
#ifndef SECTION
# define SECTION
#endif
extern const union
{
int4 i[880];
double x[440];
} __sincostab attribute_hidden;
static const double
sn3 = -1.66666666666664880952546298448555E-01,
sn5 = 8.33333214285722277379541354343671E-03,
cs2 = 4.99999999999999999999950396842453E-01,
cs4 = -4.16666666666664434524222570944589E-02,
cs6 = 1.38888874007937613028114285595617E-03;
int __branred (double x, double *a, double *aa);
/* Given a number partitioned into X and DX, this function computes the cosine
of the number by combining the sin and cos of X (as computed by a variation
of the Taylor series) with the values looked up from the sin/cos table to
get the result. */
static __always_inline double
do_cos (double x, double dx)
{
mynumber u;
if (x < 0)
dx = -dx;
u.x = big + fabs (x);
x = fabs (x) - (u.x - big) + dx;
double xx, s, sn, ssn, c, cs, ccs, cor;
xx = x * x;
s = x + x * xx * (sn3 + xx * sn5);
c = xx * (cs2 + xx * (cs4 + xx * cs6));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
cor = (ccs - s * ssn - cs * c) - sn * s;
return cs + cor;
}
/* Given a number partitioned into X and DX, this function computes the sine of
the number by combining the sin and cos of X (as computed by a variation of
the Taylor series) with the values looked up from the sin/cos table to get
the result. */
static __always_inline double
do_sin (double x, double dx)
{
double xold = x;
/* Max ULP is 0.501 if |x| < 0.126, otherwise ULP is 0.518. */
if (fabs (x) < 0.126)
return TAYLOR_SIN (x * x, x, dx);
mynumber u;
if (x <= 0)
dx = -dx;
u.x = big + fabs (x);
x = fabs (x) - (u.x - big);
double xx, s, sn, ssn, c, cs, ccs, cor;
xx = x * x;
s = x + (dx + x * xx * (sn3 + xx * sn5));
c = x * dx + xx * (cs2 + xx * (cs4 + xx * cs6));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
cor = (ssn + s * ccs - sn * c) + cs * s;
return copysign (sn + cor, xold);
}
/* Reduce range of x to within PI/2 with abs (x) < 105414350. The high part
is written to *a, the low part to *da. Range reduction is accurate to 136
bits so that when x is large and *a very close to zero, all 53 bits of *a
are correct. */
static __always_inline int4
reduce_sincos (double x, double *a, double *da)
{
mynumber v;
double t = (x * hpinv + toint);
double xn = t - toint;
v.x = t;
double y = (x - xn * mp1) - xn * mp2;
int4 n = v.i[LOW_HALF] & 3;
double b, db, t1, t2;
t1 = xn * pp3;
t2 = y - t1;
db = (y - t2) - t1;
t1 = xn * pp4;
b = t2 - t1;
db += (t2 - b) - t1;
*a = b;
*da = db;
return n;
}
/* Compute sin or cos (A + DA) for the given quadrant N. */
static __always_inline double
do_sincos (double a, double da, int4 n)
{
double retval;
if (n & 1)
/* Max ULP is 0.513. */
retval = do_cos (a, da);
else
/* Max ULP is 0.501 if xx < 0.01588, otherwise ULP is 0.518. */
retval = do_sin (a, da);
return (n & 2) ? -retval : retval;
}
/*******************************************************************/
/* An ultimate sin routine. Given an IEEE double machine number x */
/* it computes the rounded value of sin(x). */
/*******************************************************************/
#ifndef IN_SINCOS
double
SECTION
__sin (double x)
{
double t, a, da;
mynumber u;
int4 k, m, n;
double retval = 0;
SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
u.x = x;
m = u.i[HIGH_HALF];
k = 0x7fffffff & m; /* no sign */
if (k < 0x3e500000) /* if x->0 =>sin(x)=x */
{
math_check_force_underflow (x);
retval = x;
}
/*--------------------------- 2^-26<|x|< 0.855469---------------------- */
else if (k < 0x3feb6000)
{
/* Max ULP is 0.548. */
retval = do_sin (x, 0);
} /* else if (k < 0x3feb6000) */
/*----------------------- 0.855469 <|x|<2.426265 ----------------------*/
else if (k < 0x400368fd)
{
t = hp0 - fabs (x);
/* Max ULP is 0.51. */
retval = copysign (do_cos (t, hp1), x);
} /* else if (k < 0x400368fd) */
/*-------------------------- 2.426265<|x|< 105414350 ----------------------*/
else if (k < 0x419921FB)
{
n = reduce_sincos (x, &a, &da);
retval = do_sincos (a, da, n);
} /* else if (k < 0x419921FB ) */
/* --------------------105414350 <|x| <2^1024------------------------------*/
else if (k < 0x7ff00000)
{
n = __branred (x, &a, &da);
retval = do_sincos (a, da, n);
}
/*--------------------- |x| > 2^1024 ----------------------------------*/
else
{
if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
__set_errno (EDOM);
retval = x / x;
}
return retval;
}
/*******************************************************************/
/* An ultimate cos routine. Given an IEEE double machine number x */
/* it computes the rounded value of cos(x). */
/*******************************************************************/
double
SECTION
__cos (double x)
{
double y, a, da;
mynumber u;
int4 k, m, n;
double retval = 0;
SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
u.x = x;
m = u.i[HIGH_HALF];
k = 0x7fffffff & m;
/* |x|<2^-27 => cos(x)=1 */
if (k < 0x3e400000)
retval = 1.0;
else if (k < 0x3feb6000)
{ /* 2^-27 < |x| < 0.855469 */
/* Max ULP is 0.51. */
retval = do_cos (x, 0);
} /* else if (k < 0x3feb6000) */
else if (k < 0x400368fd)
{ /* 0.855469 <|x|<2.426265 */ ;
y = hp0 - fabs (x);
a = y + hp1;
da = (y - a) + hp1;
/* Max ULP is 0.501 if xx < 0.01588 or 0.518 otherwise.
Range reduction uses 106 bits here which is sufficient. */
retval = do_sin (a, da);
} /* else if (k < 0x400368fd) */
else if (k < 0x419921FB)
{ /* 2.426265<|x|< 105414350 */
n = reduce_sincos (x, &a, &da);
retval = do_sincos (a, da, n + 1);
} /* else if (k < 0x419921FB ) */
/* 105414350 <|x| <2^1024 */
else if (k < 0x7ff00000)
{
n = __branred (x, &a, &da);
retval = do_sincos (a, da, n + 1);
}
else
{
if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
__set_errno (EDOM);
retval = x / x; /* |x| > 2^1024 */
}
return retval;
}
#ifndef __cos
libm_alias_double (__cos, cos)
#endif
#ifndef __sin
libm_alias_double (__sin, sin)
#endif
#endif