2 * Copyright (c) 2008 David Schultz <das@FreeBSD.org>
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
28 * Tests for corner cases in the inverse trigonometric functions. Some
29 * accuracy tests are included as well, but these are very basic
30 * sanity checks, not intended to be comprehensive.
33 #include <sys/cdefs.h>
34 __FBSDID("$FreeBSD$");
42 #define ALL_STD_EXCEPT (FE_DIVBYZERO | FE_INEXACT | FE_INVALID | \
43 FE_OVERFLOW | FE_UNDERFLOW)
45 #define LEN(a) (sizeof(a) / sizeof((a)[0]))
47 #pragma STDC FENV_ACCESS ON
50 * Test that a function returns the correct value and sets the
51 * exception flags correctly. A tolerance specifying the maximum
52 * relative error allowed may be specified. For the 'testall'
53 * functions, the tolerance is specified in ulps.
55 * These are macros instead of functions so that assert provides more
56 * meaningful error messages.
58 #define test_tol(func, x, result, tol, excepts) do { \
59 volatile long double _in = (x), _out = (result); \
60 assert(feclearexcept(FE_ALL_EXCEPT) == 0); \
61 assert(fpequal(func(_in), _out, (tol))); \
62 assert((func, fetestexcept(ALL_STD_EXCEPT) == (excepts))); \
64 #define test(func, x, result, excepts) \
65 test_tol(func, (x), (result), 0, (excepts))
67 #define testall_tol(prefix, x, result, tol, excepts) do { \
68 test_tol(prefix, (double)(x), (double)(result), \
69 (tol) * ldexp(1.0, 1 - DBL_MANT_DIG), (excepts)); \
70 test_tol(prefix##f, (float)(x), (float)(result), \
71 (tol) * ldexpf(1.0, 1 - FLT_MANT_DIG), (excepts)); \
72 test_tol(prefix##l, (x), (result), \
73 (tol) * ldexpl(1.0, 1 - LDBL_MANT_DIG), (excepts)); \
75 #define testall(prefix, x, result, excepts) \
76 testall_tol(prefix, (x), (result), 0, (excepts))
78 #define test2_tol(func, y, x, result, tol, excepts) do { \
79 volatile long double _iny = (y), _inx = (x), _out = (result); \
80 assert(feclearexcept(FE_ALL_EXCEPT) == 0); \
81 assert(fpequal(func(_iny, _inx), _out, (tol))); \
82 assert((func, fetestexcept(ALL_STD_EXCEPT) == (excepts))); \
84 #define test2(func, y, x, result, excepts) \
85 test2_tol(func, (y), (x), (result), 0, (excepts))
87 #define testall2_tol(prefix, y, x, result, tol, excepts) do { \
88 test2_tol(prefix, (double)(y), (double)(x), (double)(result), \
89 (tol) * ldexp(1.0, 1 - DBL_MANT_DIG), (excepts)); \
90 test2_tol(prefix##f, (float)(y), (float)(x), (float)(result), \
91 (tol) * ldexpf(1.0, 1 - FLT_MANT_DIG), (excepts)); \
92 test2_tol(prefix##l, (y), (x), (result), \
93 (tol) * ldexpl(1.0, 1 - LDBL_MANT_DIG), (excepts)); \
95 #define testall2(prefix, y, x, result, excepts) \
96 testall2_tol(prefix, (y), (x), (result), 0, (excepts))
99 pi = 3.14159265358979323846264338327950280e+00L,
100 pio3 = 1.04719755119659774615421446109316766e+00L,
101 c3pi = 9.42477796076937971538793014983850839e+00L,
102 c5pi = 1.57079632679489661923132169163975140e+01L,
103 c7pi = 2.19911485751285526692385036829565196e+01L,
104 c5pio3 = 5.23598775598298873077107230546583851e+00L,
105 sqrt2m1 = 4.14213562373095048801688724209698081e-01L;
108 * Determine whether x and y are equal to within a relative error of tol,
109 * with two special rules:
114 fpequal(long double x, long double y, long double tol)
119 if (isnan(x) && isnan(y))
121 if (signbit(x) != signbit(y))
128 /* Hard case: need to check the tolerance. */
130 ret = fabsl(x - y) <= fabsl(y * tol);
136 * Test special case inputs in asin(), acos() and atan(): signed
137 * zeroes, infinities, and NaNs.
143 testall(asin, 0.0, 0.0, 0);
144 testall(acos, 0.0, pi / 2, FE_INEXACT);
145 testall(atan, 0.0, 0.0, 0);
146 testall(asin, -0.0, -0.0, 0);
147 testall(acos, -0.0, pi / 2, FE_INEXACT);
148 testall(atan, -0.0, -0.0, 0);
150 testall(asin, INFINITY, NAN, FE_INVALID);
151 testall(acos, INFINITY, NAN, FE_INVALID);
152 testall(atan, INFINITY, pi / 2, FE_INEXACT);
153 testall(asin, -INFINITY, NAN, FE_INVALID);
154 testall(acos, -INFINITY, NAN, FE_INVALID);
155 testall(atan, -INFINITY, -pi / 2, FE_INEXACT);
157 testall(asin, NAN, NAN, 0);
158 testall(acos, NAN, NAN, 0);
159 testall(atan, NAN, NAN, 0);
163 * Test special case inputs in atan2(), where the exact value of y/x is
164 * zero or non-finite.
167 test_special_atan2(void)
172 testall2(atan2, 0.0, -0.0, pi, FE_INEXACT);
173 testall2(atan2, -0.0, -0.0, -pi, FE_INEXACT);
174 testall2(atan2, 0.0, 0.0, 0.0, 0);
175 testall2(atan2, -0.0, 0.0, -0.0, 0);
177 testall2(atan2, INFINITY, -INFINITY, c3pi / 4, FE_INEXACT);
178 testall2(atan2, -INFINITY, -INFINITY, -c3pi / 4, FE_INEXACT);
179 testall2(atan2, INFINITY, INFINITY, pi / 4, FE_INEXACT);
180 testall2(atan2, -INFINITY, INFINITY, -pi / 4, FE_INEXACT);
182 /* Tests with one input in the range (0, Inf]. */
184 for (e = FLT_MIN_EXP - FLT_MANT_DIG; e <= FLT_MAX_EXP; e++) {
185 test2(atan2f, 0.0, ldexpf(z, e), 0.0, 0);
186 test2(atan2f, -0.0, ldexpf(z, e), -0.0, 0);
187 test2(atan2f, 0.0, ldexpf(-z, e), (float)pi, FE_INEXACT);
188 test2(atan2f, -0.0, ldexpf(-z, e), (float)-pi, FE_INEXACT);
189 test2(atan2f, ldexpf(z, e), 0.0, (float)pi / 2, FE_INEXACT);
190 test2(atan2f, ldexpf(z, e), -0.0, (float)pi / 2, FE_INEXACT);
191 test2(atan2f, ldexpf(-z, e), 0.0, (float)-pi / 2, FE_INEXACT);
192 test2(atan2f, ldexpf(-z, e), -0.0, (float)-pi / 2, FE_INEXACT);
194 for (e = DBL_MIN_EXP - DBL_MANT_DIG; e <= DBL_MAX_EXP; e++) {
195 test2(atan2, 0.0, ldexp(z, e), 0.0, 0);
196 test2(atan2, -0.0, ldexp(z, e), -0.0, 0);
197 test2(atan2, 0.0, ldexp(-z, e), (double)pi, FE_INEXACT);
198 test2(atan2, -0.0, ldexp(-z, e), (double)-pi, FE_INEXACT);
199 test2(atan2, ldexp(z, e), 0.0, (double)pi / 2, FE_INEXACT);
200 test2(atan2, ldexp(z, e), -0.0, (double)pi / 2, FE_INEXACT);
201 test2(atan2, ldexp(-z, e), 0.0, (double)-pi / 2, FE_INEXACT);
202 test2(atan2, ldexp(-z, e), -0.0, (double)-pi / 2, FE_INEXACT);
204 for (e = LDBL_MIN_EXP - LDBL_MANT_DIG; e <= LDBL_MAX_EXP; e++) {
205 test2(atan2l, 0.0, ldexpl(z, e), 0.0, 0);
206 test2(atan2l, -0.0, ldexpl(z, e), -0.0, 0);
207 test2(atan2l, 0.0, ldexpl(-z, e), pi, FE_INEXACT);
208 test2(atan2l, -0.0, ldexpl(-z, e), -pi, FE_INEXACT);
209 test2(atan2l, ldexpl(z, e), 0.0, pi / 2, FE_INEXACT);
210 test2(atan2l, ldexpl(z, e), -0.0, pi / 2, FE_INEXACT);
211 test2(atan2l, ldexpl(-z, e), 0.0, -pi / 2, FE_INEXACT);
212 test2(atan2l, ldexpl(-z, e), -0.0, -pi / 2, FE_INEXACT);
215 /* Tests with one input in the range (0, Inf). */
216 for (e = FLT_MIN_EXP - FLT_MANT_DIG; e <= FLT_MAX_EXP - 1; e++) {
217 test2(atan2f, ldexpf(z, e), INFINITY, 0.0, 0);
218 test2(atan2f, ldexpf(-z,e), INFINITY, -0.0, 0);
219 test2(atan2f, ldexpf(z, e), -INFINITY, (float)pi, FE_INEXACT);
220 test2(atan2f, ldexpf(-z,e), -INFINITY, (float)-pi, FE_INEXACT);
221 test2(atan2f, INFINITY, ldexpf(z,e), (float)pi/2, FE_INEXACT);
222 test2(atan2f, INFINITY, ldexpf(-z,e), (float)pi/2, FE_INEXACT);
223 test2(atan2f, -INFINITY, ldexpf(z,e), (float)-pi/2,FE_INEXACT);
224 test2(atan2f, -INFINITY, ldexpf(-z,e),(float)-pi/2,FE_INEXACT);
226 for (e = DBL_MIN_EXP - DBL_MANT_DIG; e <= DBL_MAX_EXP - 1; e++) {
227 test2(atan2, ldexp(z, e), INFINITY, 0.0, 0);
228 test2(atan2, ldexp(-z,e), INFINITY, -0.0, 0);
229 test2(atan2, ldexp(z, e), -INFINITY, (double)pi, FE_INEXACT);
230 test2(atan2, ldexp(-z,e), -INFINITY, (double)-pi, FE_INEXACT);
231 test2(atan2, INFINITY, ldexp(z,e), (double)pi/2, FE_INEXACT);
232 test2(atan2, INFINITY, ldexp(-z,e), (double)pi/2, FE_INEXACT);
233 test2(atan2, -INFINITY, ldexp(z,e), (double)-pi/2,FE_INEXACT);
234 test2(atan2, -INFINITY, ldexp(-z,e),(double)-pi/2,FE_INEXACT);
236 for (e = LDBL_MIN_EXP - LDBL_MANT_DIG; e <= LDBL_MAX_EXP - 1; e++) {
237 test2(atan2l, ldexpl(z, e), INFINITY, 0.0, 0);
238 test2(atan2l, ldexpl(-z,e), INFINITY, -0.0, 0);
239 test2(atan2l, ldexpl(z, e), -INFINITY, pi, FE_INEXACT);
240 test2(atan2l, ldexpl(-z,e), -INFINITY, -pi, FE_INEXACT);
241 test2(atan2l, INFINITY, ldexpl(z, e), pi / 2, FE_INEXACT);
242 test2(atan2l, INFINITY, ldexpl(-z, e), pi / 2, FE_INEXACT);
243 test2(atan2l, -INFINITY, ldexpl(z, e), -pi / 2, FE_INEXACT);
244 test2(atan2l, -INFINITY, ldexpl(-z, e), -pi / 2, FE_INEXACT);
249 * Test various inputs to asin(), acos() and atan() and verify that the
250 * results are accurate to within 1 ulp.
256 /* We expect correctly rounded results for these basic cases. */
257 testall(asin, 1.0, pi / 2, FE_INEXACT);
258 testall(acos, 1.0, 0, 0);
259 testall(atan, 1.0, pi / 4, FE_INEXACT);
260 testall(asin, -1.0, -pi / 2, FE_INEXACT);
261 testall(acos, -1.0, pi, FE_INEXACT);
262 testall(atan, -1.0, -pi / 4, FE_INEXACT);
265 * Here we expect answers to be within 1 ulp, although inexactness
266 * in the input, combined with double rounding, could cause larger
270 testall_tol(asin, sqrtl(2) / 2, pi / 4, 1, FE_INEXACT);
271 testall_tol(acos, sqrtl(2) / 2, pi / 4, 1, FE_INEXACT);
272 testall_tol(asin, -sqrtl(2) / 2, -pi / 4, 1, FE_INEXACT);
273 testall_tol(acos, -sqrtl(2) / 2, c3pi / 4, 1, FE_INEXACT);
275 testall_tol(asin, sqrtl(3) / 2, pio3, 1, FE_INEXACT);
276 testall_tol(acos, sqrtl(3) / 2, pio3 / 2, 1, FE_INEXACT);
277 testall_tol(atan, sqrtl(3), pio3, 1, FE_INEXACT);
278 testall_tol(asin, -sqrtl(3) / 2, -pio3, 1, FE_INEXACT);
279 testall_tol(acos, -sqrtl(3) / 2, c5pio3 / 2, 1, FE_INEXACT);
280 testall_tol(atan, -sqrtl(3), -pio3, 1, FE_INEXACT);
282 testall_tol(atan, sqrt2m1, pi / 8, 1, FE_INEXACT);
283 testall_tol(atan, -sqrt2m1, -pi / 8, 1, FE_INEXACT);
287 * Test inputs to atan2() where x is a power of 2. These are easy cases
288 * because y/x is exact.
294 testall2(atan2, 1.0, 1.0, pi / 4, FE_INEXACT);
295 testall2(atan2, 1.0, -1.0, c3pi / 4, FE_INEXACT);
296 testall2(atan2, -1.0, 1.0, -pi / 4, FE_INEXACT);
297 testall2(atan2, -1.0, -1.0, -c3pi / 4, FE_INEXACT);
299 testall2_tol(atan2, sqrt2m1 * 2, 2.0, pi / 8, 1, FE_INEXACT);
300 testall2_tol(atan2, sqrt2m1 * 2, -2.0, c7pi / 8, 1, FE_INEXACT);
301 testall2_tol(atan2, -sqrt2m1 * 2, 2.0, -pi / 8, 1, FE_INEXACT);
302 testall2_tol(atan2, -sqrt2m1 * 2, -2.0, -c7pi / 8, 1, FE_INEXACT);
304 testall2_tol(atan2, sqrtl(3) * 0.5, 0.5, pio3, 1, FE_INEXACT);
305 testall2_tol(atan2, sqrtl(3) * 0.5, -0.5, pio3 * 2, 1, FE_INEXACT);
306 testall2_tol(atan2, -sqrtl(3) * 0.5, 0.5, -pio3, 1, FE_INEXACT);
307 testall2_tol(atan2, -sqrtl(3) * 0.5, -0.5, -pio3 * 2, 1, FE_INEXACT);
311 * Test inputs very close to 0.
316 float tiny = 0x1.23456p-120f;
318 testall(asin, tiny, tiny, FE_INEXACT);
319 testall(acos, tiny, pi / 2, FE_INEXACT);
320 testall(atan, tiny, tiny, FE_INEXACT);
322 testall(asin, -tiny, -tiny, FE_INEXACT);
323 testall(acos, -tiny, pi / 2, FE_INEXACT);
324 testall(atan, -tiny, -tiny, FE_INEXACT);
326 /* Test inputs to atan2() that would cause y/x to underflow. */
327 test2(atan2f, 0x1.0p-100, 0x1.0p100, 0.0, FE_INEXACT | FE_UNDERFLOW);
328 test2(atan2, 0x1.0p-1000, 0x1.0p1000, 0.0, FE_INEXACT | FE_UNDERFLOW);
329 test2(atan2l, ldexpl(1.0, 100 - LDBL_MAX_EXP),
330 ldexpl(1.0, LDBL_MAX_EXP - 100), 0.0, FE_INEXACT | FE_UNDERFLOW);
331 test2(atan2f, -0x1.0p-100, 0x1.0p100, -0.0, FE_INEXACT | FE_UNDERFLOW);
332 test2(atan2, -0x1.0p-1000, 0x1.0p1000, -0.0, FE_INEXACT | FE_UNDERFLOW);
333 test2(atan2l, -ldexpl(1.0, 100 - LDBL_MAX_EXP),
334 ldexpl(1.0, LDBL_MAX_EXP - 100), -0.0, FE_INEXACT | FE_UNDERFLOW);
335 test2(atan2f, 0x1.0p-100, -0x1.0p100, (float)pi, FE_INEXACT);
336 test2(atan2, 0x1.0p-1000, -0x1.0p1000, (double)pi, FE_INEXACT);
337 test2(atan2l, ldexpl(1.0, 100 - LDBL_MAX_EXP),
338 -ldexpl(1.0, LDBL_MAX_EXP - 100), pi, FE_INEXACT);
339 test2(atan2f, -0x1.0p-100, -0x1.0p100, (float)-pi, FE_INEXACT);
340 test2(atan2, -0x1.0p-1000, -0x1.0p1000, (double)-pi, FE_INEXACT);
341 test2(atan2l, -ldexpl(1.0, 100 - LDBL_MAX_EXP),
342 -ldexpl(1.0, LDBL_MAX_EXP - 100), -pi, FE_INEXACT);
346 * Test very large inputs to atan().
351 float huge = 0x1.23456p120;
353 testall(atan, huge, pi / 2, FE_INEXACT);
354 testall(atan, -huge, -pi / 2, FE_INEXACT);
356 /* Test inputs to atan2() that would cause y/x to overflow. */
357 test2(atan2f, 0x1.0p100, 0x1.0p-100, (float)pi / 2, FE_INEXACT);
358 test2(atan2, 0x1.0p1000, 0x1.0p-1000, (double)pi / 2, FE_INEXACT);
359 test2(atan2l, ldexpl(1.0, LDBL_MAX_EXP - 100),
360 ldexpl(1.0, 100 - LDBL_MAX_EXP), pi / 2, FE_INEXACT);
361 test2(atan2f, -0x1.0p100, 0x1.0p-100, (float)-pi / 2, FE_INEXACT);
362 test2(atan2, -0x1.0p1000, 0x1.0p-1000, (double)-pi / 2, FE_INEXACT);
363 test2(atan2l, -ldexpl(1.0, LDBL_MAX_EXP - 100),
364 ldexpl(1.0, 100 - LDBL_MAX_EXP), -pi / 2, FE_INEXACT);
366 test2(atan2f, 0x1.0p100, -0x1.0p-100, (float)pi / 2, FE_INEXACT);
367 test2(atan2, 0x1.0p1000, -0x1.0p-1000, (double)pi / 2, FE_INEXACT);
368 test2(atan2l, ldexpl(1.0, LDBL_MAX_EXP - 100),
369 -ldexpl(1.0, 100 - LDBL_MAX_EXP), pi / 2, FE_INEXACT);
370 test2(atan2f, -0x1.0p100, -0x1.0p-100, (float)-pi / 2, FE_INEXACT);
371 test2(atan2, -0x1.0p1000, -0x1.0p-1000, (double)-pi / 2, FE_INEXACT);
372 test2(atan2l, -ldexpl(1.0, LDBL_MAX_EXP - 100),
373 -ldexpl(1.0, 100 - LDBL_MAX_EXP), -pi / 2, FE_INEXACT);
377 * Test that sin(asin(x)) == x, and similarly for acos() and atan().
378 * You need to have a working sinl(), cosl(), and tanl() for these
385 return (sinl(asinf(x)));
392 return (sinl(asin(x)));
396 sinasinl(long double x)
399 return (sinl(asinl(x)));
406 return (cosl(acosf(x)));
413 return (cosl(acos(x)));
417 cosacosl(long double x)
420 return (cosl(acosl(x)));
427 return (tanl(atanf(x)));
434 return (tanl(atan(x)));
438 tanatanl(long double x)
441 return (tanl(atanl(x)));
449 for (i = -1; i <= 1; i += 0x1.0p-12f) {
450 testall_tol(sinasin, i, i, 2, i == 0 ? 0 : FE_INEXACT);
451 /* The relative error for cosacos is very large near x=0. */
452 if (fabsf(i) > 0x1.0p-4f)
453 testall_tol(cosacos, i, i, 16, i == 1 ? 0 : FE_INEXACT);
454 testall_tol(tanatan, i, i, 2, i == 0 ? 0 : FE_INEXACT);
459 main(int argc, char *argv[])
465 printf("ok 1 - special\n");
467 test_special_atan2();
468 printf("ok 2 - atan2 special\n");
471 printf("ok 3 - accuracy\n");
474 printf("ok 4 - atan2 p2x\n");
477 printf("ok 5 - tiny inputs\n");
480 printf("ok 6 - atan huge inputs\n");
483 printf("ok 7 - inverse\n");