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1.1 root 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) 8/22/85";
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:
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: }
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:
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:
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:
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: }
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