1 /*
2 * Copyright (c) 1985 Regents of the University of California.
3 *
4 * Use and reproduction of this software are granted in accordance with
5 * the terms and conditions specified in the Berkeley Software License
6 * Agreement (in particular, this entails acknowledgement of the programs'
7 * source, and inclusion of this notice) with the additional understanding
8 * that all recipients should regard themselves as participants in an
9 * ongoing research project and hence should feel obligated to report
10 * their experiences (good or bad) with these elementary function codes,
11 * using "sendbug 4bsd-bugs@BERKELEY", to the authors.
12 */
13
14 #ifndef lint
15 static char sccsid[] = "@(#)trig.c 1.2 (Berkeley) 08/22/85";
16 #endif not lint
17
18 /* SIN(X), COS(X), TAN(X)
19 * RETURN THE SINE, COSINE, AND TANGENT OF X RESPECTIVELY
20 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
21 * CODED IN C BY K.C. NG, 1/8/85;
22 * REVISED BY W. Kahan and K.C. NG, 8/17/85.
23 *
24 * Required system supported functions:
25 * copysign(x,y)
26 * finite(x)
27 * drem(x,p)
28 *
29 * Static kernel functions:
30 * sin__S(z) ....sin__S(x*x) return (sin(x)-x)/x
31 * cos__C(z) ....cos__C(x*x) return cos(x)-1-x*x/2
32 *
33 * Method.
34 * Let S and C denote the polynomial approximations to sin and cos
35 * respectively on [-PI/4, +PI/4].
36 *
37 * SIN and COS:
38 * 1. Reduce the argument into [-PI , +PI] by the remainder function.
39 * 2. For x in (-PI,+PI), there are three cases:
40 * case 1: |x| < PI/4
41 * case 2: PI/4 <= |x| < 3PI/4
42 * case 3: 3PI/4 <= |x|.
43 * SIN and COS of x are computed by:
44 *
45 * sin(x) cos(x) remark
46 * ----------------------------------------------------------
47 * case 1 S(x) C(x)
48 * case 2 sign(x)*C(y) S(y) y=PI/2-|x|
49 * case 3 S(y) -C(y) y=sign(x)*(PI-|x|)
50 * ----------------------------------------------------------
51 *
52 * TAN:
53 * 1. Reduce the argument into [-PI/2 , +PI/2] by the remainder function.
54 * 2. For x in (-PI/2,+PI/2), there are two cases:
55 * case 1: |x| < PI/4
56 * case 2: PI/4 <= |x| < PI/2
57 * TAN of x is computed by:
58 *
59 * tan (x) remark
60 * ----------------------------------------------------------
61 * case 1 S(x)/C(x)
62 * case 2 C(y)/S(y) y=sign(x)*(PI/2-|x|)
63 * ----------------------------------------------------------
64 *
65 * Notes:
66 * 1. S(y) and C(y) were computed by:
67 * S(y) = y+y*sin__S(y*y)
68 * C(y) = 1-(y*y/2-cos__C(x*x)) ... if y*y/2 < thresh,
69 * = 0.5-((y*y/2-0.5)-cos__C(x*x)) ... if y*y/2 >= thresh.
70 * where
71 * thresh = 0.5*(acos(3/4)**2)
72 *
73 * 2. For better accuracy, we use the following formula for S/C for tan
74 * (k=0): let ss=sin__S(y*y), and cc=cos__C(y*y), then
75 *
76 * y+y*ss (y*y/2-cc)+ss
77 * S(y)/C(y) = -------- = y + y * ---------------.
78 * C C
79 *
80 *
81 * Special cases:
82 * Let trig be any of sin, cos, or tan.
83 * trig(+-INF) is NaN, with signals;
84 * trig(NaN) is that NaN;
85 * trig(n*PI/2) is exact for any integer n, provided n*PI is
86 * representable; otherwise, trig(x) is inexact.
87 *
88 * Accuracy:
89 * trig(x) returns the exact trig(x*pi/PI) nearly rounded, where
90 *
91 * Decimal:
92 * pi = 3.141592653589793 23846264338327 .....
93 * 53 bits PI = 3.141592653589793 115997963 ..... ,
94 * 56 bits PI = 3.141592653589793 227020265 ..... ,
95 *
96 * Hexadecimal:
97 * pi = 3.243F6A8885A308D313198A2E....
98 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
99 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
100 *
101 * In a test run with 1,024,000 random arguments on a VAX, the maximum
102 * observed errors (compared with the exact trig(x*pi/PI)) were
103 * tan(x) : 2.09 ulps (around 4.716340404662354)
104 * sin(x) : .861 ulps
105 * cos(x) : .857 ulps
106 *
107 * Constants:
108 * The hexadecimal values are the intended ones for the following constants.
109 * The decimal values may be used, provided that the compiler will convert
110 * from decimal to binary accurately enough to produce the hexadecimal values
111 * shown.
112 */
113
114 #ifdef VAX
115 /*thresh = 2.6117239648121182150E-1 , Hex 2^ -1 * .85B8636B026EA0 */
116 /*PIo4 = 7.8539816339744830676E-1 , Hex 2^ 0 * .C90FDAA22168C2 */
117 /*PIo2 = 1.5707963267948966135E0 , Hex 2^ 1 * .C90FDAA22168C2 */
118 /*PI3o4 = 2.3561944901923449203E0 , Hex 2^ 2 * .96CBE3F9990E92 */
119 /*PI = 3.1415926535897932270E0 , Hex 2^ 2 * .C90FDAA22168C2 */
120 /*PI2 = 6.2831853071795864540E0 ; Hex 2^ 3 * .C90FDAA22168C2 */
121 static long threshx[] = { 0xb8633f85, 0x6ea06b02};
122 #define thresh (*(double*)threshx)
123 static long PIo4x[] = { 0x0fda4049, 0x68c2a221};
124 #define PIo4 (*(double*)PIo4x)
125 static long PIo2x[] = { 0x0fda40c9, 0x68c2a221};
126 #define PIo2 (*(double*)PIo2x)
127 static long PI3o4x[] = { 0xcbe34116, 0x0e92f999};
128 #define PI3o4 (*(double*)PI3o4x)
129 static long PIx[] = { 0x0fda4149, 0x68c2a221};
130 #define PI (*(double*)PIx)
131 static long PI2x[] = { 0x0fda41c9, 0x68c2a221};
132 #define PI2 (*(double*)PI2x)
133 #else /* IEEE double */
134 static double
135 thresh = 2.6117239648121182150E-1 , /*Hex 2^ -2 * 1.0B70C6D604DD4 */
136 PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */
137 PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */
138 PI3o4 = 2.3561944901923448370E0 , /*Hex 2^ 1 * 1.2D97C7F3321D2 */
139 PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */
140 PI2 = 6.2831853071795862320E0 ; /*Hex 2^ 2 * 1.921FB54442D18 */
141 #endif
142 static double zero=0, one=1, negone= -1, half=1.0/2.0,
143 small=1E-10, /* 1+small**2==1; better values for small:
144 small = 1.5E-9 for VAX D
145 = 1.2E-8 for IEEE Double
146 = 2.8E-10 for IEEE Extended */
147 big=1E20; /* big = 1/(small**2) */
148
tan(x)149 double tan(x)
150 double x;
151 {
152 double copysign(),drem(),cos__C(),sin__S(),a,z,ss,cc,c;
153 int finite(),k;
154
155 /* tan(NaN) and tan(INF) must be NaN */
156 if(!finite(x)) return(x-x);
157 x=drem(x,PI); /* reduce x into [-PI/2, PI/2] */
158 a=copysign(x,one); /* ... = abs(x) */
159 if ( a >= PIo4 ) {k=1; x = copysign( PIo2 - a , x ); }
160 else { k=0; if(a < small ) { big + a; return(x); }}
161
162 z = x*x;
163 cc = cos__C(z);
164 ss = sin__S(z);
165 z = z*half ; /* Next get c = cos(x) accurately */
166 c = (z >= thresh )? half-((z-half)-cc) : one-(z-cc);
167 if (k==0) return ( x + (x*(z-(cc-ss)))/c ); /* sin/cos */
168 return( c/(x+x*ss) ); /* ... cos/sin */
169
170
171 }
sin(x)172 double sin(x)
173 double x;
174 {
175 double copysign(),drem(),sin__S(),cos__C(),a,c,z;
176 int finite();
177
178 /* sin(NaN) and sin(INF) must be NaN */
179 if(!finite(x)) return(x-x);
180 x=drem(x,PI2); /* reduce x into [-PI, PI] */
181 a=copysign(x,one);
182 if( a >= PIo4 ) {
183 if( a >= PI3o4 ) /* .. in [3PI/4, PI ] */
184 x=copysign((a=PI-a),x);
185
186 else { /* .. in [PI/4, 3PI/4] */
187 a=PIo2-a; /* return sign(x)*C(PI/2-|x|) */
188 z=a*a;
189 c=cos__C(z);
190 z=z*half;
191 a=(z>=thresh)?half-((z-half)-c):one-(z-c);
192 return(copysign(a,x));
193 }
194 }
195
196 /* return S(x) */
197 if( a < small) { big + a; return(x);}
198 return(x+x*sin__S(x*x));
199 }
200
cos(x)201 double cos(x)
202 double x;
203 {
204 double copysign(),drem(),sin__S(),cos__C(),a,c,z,s=1.0;
205 int finite();
206
207 /* cos(NaN) and cos(INF) must be NaN */
208 if(!finite(x)) return(x-x);
209 x=drem(x,PI2); /* reduce x into [-PI, PI] */
210 a=copysign(x,one);
211 if ( a >= PIo4 ) {
212 if ( a >= PI3o4 ) /* .. in [3PI/4, PI ] */
213 { a=PI-a; s= negone; }
214
215 else /* .. in [PI/4, 3PI/4] */
216 /* return S(PI/2-|x|) */
217 { a=PIo2-a; return(a+a*sin__S(a*a));}
218 }
219
220
221 /* return s*C(a) */
222 if( a < small) { big + a; return(s);}
223 z=a*a;
224 c=cos__C(z);
225 z=z*half;
226 a=(z>=thresh)?half-((z-half)-c):one-(z-c);
227 return(copysign(a,s));
228 }
229
230
231 /* sin__S(x*x)
232 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
233 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X)
234 * CODED IN C BY K.C. NG, 1/21/85;
235 * REVISED BY K.C. NG on 8/13/85.
236 *
237 * sin(x*k) - x
238 * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded
239 * x
240 * value of pi in machine precision:
241 *
242 * Decimal:
243 * pi = 3.141592653589793 23846264338327 .....
244 * 53 bits PI = 3.141592653589793 115997963 ..... ,
245 * 56 bits PI = 3.141592653589793 227020265 ..... ,
246 *
247 * Hexadecimal:
248 * pi = 3.243F6A8885A308D313198A2E....
249 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18
250 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2
251 *
252 * Method:
253 * 1. Let z=x*x. Create a polynomial approximation to
254 * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5).
255 * Then
256 * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5)
257 *
258 * The coefficient S's are obtained by a special Remez algorithm.
259 *
260 * Accuracy:
261 * In the absence of rounding error, the approximation has absolute error
262 * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE.
263 *
264 * Constants:
265 * The hexadecimal values are the intended ones for the following constants.
266 * The decimal values may be used, provided that the compiler will convert
267 * from decimal to binary accurately enough to produce the hexadecimal values
268 * shown.
269 *
270 */
271
272 #ifdef VAX
273 /*S0 = -1.6666666666666646660E-1 , Hex 2^ -2 * -.AAAAAAAAAAAA71 */
274 /*S1 = 8.3333333333297230413E-3 , Hex 2^ -6 * .8888888888477F */
275 /*S2 = -1.9841269838362403710E-4 , Hex 2^-12 * -.D00D00CF8A1057 */
276 /*S3 = 2.7557318019967078930E-6 , Hex 2^-18 * .B8EF1CA326BEDC */
277 /*S4 = -2.5051841873876551398E-8 , Hex 2^-25 * -.D73195374CE1D3 */
278 /*S5 = 1.6028995389845827653E-10 , Hex 2^-32 * .B03D9C6D26CCCC */
279 /*S6 = -6.2723499671769283121E-13 ; Hex 2^-40 * -.B08D0B7561EA82 */
280 static long S0x[] = { 0xaaaabf2a, 0xaa71aaaa};
281 #define S0 (*(double*)S0x)
282 static long S1x[] = { 0x88883d08, 0x477f8888};
283 #define S1 (*(double*)S1x)
284 static long S2x[] = { 0x0d00ba50, 0x1057cf8a};
285 #define S2 (*(double*)S2x)
286 static long S3x[] = { 0xef1c3738, 0xbedca326};
287 #define S3 (*(double*)S3x)
288 static long S4x[] = { 0x3195b3d7, 0xe1d3374c};
289 #define S4 (*(double*)S4x)
290 static long S5x[] = { 0x3d9c3030, 0xcccc6d26};
291 #define S5 (*(double*)S5x)
292 static long S6x[] = { 0x8d0bac30, 0xea827561};
293 #define S6 (*(double*)S6x)
294 #else /* IEEE double */
295 static double
296 S0 = -1.6666666666666463126E-1 , /*Hex 2^ -3 * -1.555555555550C */
297 S1 = 8.3333333332992771264E-3 , /*Hex 2^ -7 * 1.111111110C461 */
298 S2 = -1.9841269816180999116E-4 , /*Hex 2^-13 * -1.A01A019746345 */
299 S3 = 2.7557309793219876880E-6 , /*Hex 2^-19 * 1.71DE3209CDCD9 */
300 S4 = -2.5050225177523807003E-8 , /*Hex 2^-26 * -1.AE5C0E319A4EF */
301 S5 = 1.5868926979889205164E-10 ; /*Hex 2^-33 * 1.5CF61DF672B13 */
302 #endif
303
sin__S(z)304 static double sin__S(z)
305 double z;
306 {
307 #ifdef VAX
308 return(z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6)))))));
309 #else /* IEEE double */
310 return(z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5))))));
311 #endif
312 }
313
314
315 /* cos__C(x*x)
316 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS)
317 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X)
318 * CODED IN C BY K.C. NG, 1/21/85;
319 * REVISED BY K.C. NG on 8/13/85.
320 *
321 * x*x
322 * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI,
323 * 2
324 * PI is the rounded value of pi in machine precision :
325 *
326 * Decimal:
327 * pi = 3.141592653589793 23846264338327 .....
328 * 53 bits PI = 3.141592653589793 115997963 ..... ,
329 * 56 bits PI = 3.141592653589793 227020265 ..... ,
330 *
331 * Hexadecimal:
332 * pi = 3.243F6A8885A308D313198A2E....
333 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18
334 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2
335 *
336 *
337 * Method:
338 * 1. Let z=x*x. Create a polynomial approximation to
339 * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5)
340 * then
341 * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5)
342 *
343 * The coefficient C's are obtained by a special Remez algorithm.
344 *
345 * Accuracy:
346 * In the absence of rounding error, the approximation has absolute error
347 * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE.
348 *
349 *
350 * Constants:
351 * The hexadecimal values are the intended ones for the following constants.
352 * The decimal values may be used, provided that the compiler will convert
353 * from decimal to binary accurately enough to produce the hexadecimal values
354 * shown.
355 *
356 */
357
358 #ifdef VAX
359 /*C0 = 4.1666666666666504759E-2 , Hex 2^ -4 * .AAAAAAAAAAA9F0 */
360 /*C1 = -1.3888888888865302059E-3 , Hex 2^ -9 * -.B60B60B60A0CCA */
361 /*C2 = 2.4801587285601038265E-5 , Hex 2^-15 * .D00D00CDCD098F */
362 /*C3 = -2.7557313470902390219E-7 , Hex 2^-21 * -.93F27BB593E805 */
363 /*C4 = 2.0875623401082232009E-9 , Hex 2^-28 * .8F74C8FA1E3FF0 */
364 /*C5 = -1.1355178117642986178E-11 ; Hex 2^-36 * -.C7C32D0A5C5A63 */
365 static long C0x[] = { 0xaaaa3e2a, 0xa9f0aaaa};
366 #define C0 (*(double*)C0x)
367 static long C1x[] = { 0x0b60bbb6, 0x0ccab60a};
368 #define C1 (*(double*)C1x)
369 static long C2x[] = { 0x0d0038d0, 0x098fcdcd};
370 #define C2 (*(double*)C2x)
371 static long C3x[] = { 0xf27bb593, 0xe805b593};
372 #define C3 (*(double*)C3x)
373 static long C4x[] = { 0x74c8320f, 0x3ff0fa1e};
374 #define C4 (*(double*)C4x)
375 static long C5x[] = { 0xc32dae47, 0x5a630a5c};
376 #define C5 (*(double*)C5x)
377 #else /* IEEE double */
378 static double
379 C0 = 4.1666666666666504759E-2 , /*Hex 2^ -5 * 1.555555555553E */
380 C1 = -1.3888888888865301516E-3 , /*Hex 2^-10 * -1.6C16C16C14199 */
381 C2 = 2.4801587269650015769E-5 , /*Hex 2^-16 * 1.A01A01971CAEB */
382 C3 = -2.7557304623183959811E-7 , /*Hex 2^-22 * -1.27E4F1314AD1A */
383 C4 = 2.0873958177697780076E-9 , /*Hex 2^-29 * 1.1EE3B60DDDC8C */
384 C5 = -1.1250289076471311557E-11 ; /*Hex 2^-37 * -1.8BD5986B2A52E */
385 #endif
386
cos__C(z)387 static double cos__C(z)
388 double z;
389 {
390 return(z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5))))));
391 }
392