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mingw-w64: Use approximations for catanh if real part is small (bug #62332). * src/mingw-w64-complex-inverse-trig.patch: Use approximations for catanh if real part of input is small. Fix thinko in patch for casinh.
author Markus Mützel <markus.muetzel@gmx.de>
date Mon, 08 Aug 2022 20:21:02 +0200
parents 6f3c099c0d38
children dbc29160e2b2
comparison
equal deleted inserted replaced
6356:6f3c099c0d38 6357:dc5ad8056086
118 118
119 return ret; 119 return ret;
120 -- 120 --
121 2.35.3.windows.1 121 2.35.3.windows.1
122 122
123 From 98d5b204fe689ac35d348a1742723ec561a2c5f7 Mon Sep 17 00:00:00 2001 123 From cddd2eb87270f368d18f79a1070a36e4dc5464eb Mon Sep 17 00:00:00 2001
124 From: =?UTF-8?q?Markus=20M=C3=BCtzel?= <markus.muetzel@gmx.de> 124 From: =?UTF-8?q?Markus=20M=C3=BCtzel?= <markus.muetzel@gmx.de>
125 Date: Sun, 7 Aug 2022 13:21:29 +0200 125 Date: Mon, 8 Aug 2022 18:01:32 +0200
126 Subject: [PATCH 1/2] casinh: Use symmetries also for default case. 126 Subject: [PATCH 1/3] casinh: Use symmetries also for default case.
127 127
128 --- 128 ---
129 mingw-w64-crt/complex/casinh.def.h | 46 ++++++++++++++++-------------- 129 mingw-w64-crt/complex/casinh.def.h | 47 +++++++++++++++++-------------
130 1 file changed, 25 insertions(+), 21 deletions(-) 130 1 file changed, 26 insertions(+), 21 deletions(-)
131 131
132 diff --git a/mingw-w64-crt/complex/casinh.def.h b/mingw-w64-crt/complex/casinh.def.h 132 diff --git a/mingw-w64-crt/complex/casinh.def.h b/mingw-w64-crt/complex/casinh.def.h
133 index cce2d864e..d309e6392 100644 133 index cce2d864e..40ff67579 100644
134 --- a/mingw-w64-crt/complex/casinh.def.h 134 --- a/mingw-w64-crt/complex/casinh.def.h
135 +++ b/mingw-w64-crt/complex/casinh.def.h 135 +++ b/mingw-w64-crt/complex/casinh.def.h
136 @@ -89,32 +89,36 @@ __FLT_ABI(casinh) (__FLT_TYPE __complex__ z) 136 @@ -47,6 +47,7 @@ __FLT_ABI(casinh) (__FLT_TYPE __complex__ z)
137 {
138 __complex__ __FLT_TYPE ret;
139 __complex__ __FLT_TYPE x;
140 + __FLT_TYPE arz, aiz;
141 int r_class = fpclassify (__real__ z);
142 int i_class = fpclassify (__imag__ z);
143
144 @@ -89,32 +90,36 @@ __FLT_ABI(casinh) (__FLT_TYPE __complex__ z)
137 145
138 /* casinh(z) = log(z + sqrt(z*z + 1)) */ 146 /* casinh(z) = log(z + sqrt(z*z + 1)) */
139 147
140 - if (__FLT_ABI(fabs) (__real__ z) >= __FLT_CST(1.0)/__FLT_EPSILON 148 - if (__FLT_ABI(fabs) (__real__ z) >= __FLT_CST(1.0)/__FLT_EPSILON
141 - || __FLT_ABI(fabs) (__imag__ z) >= __FLT_CST(1.0)/__FLT_EPSILON) 149 - || __FLT_ABI(fabs) (__imag__ z) >= __FLT_CST(1.0)/__FLT_EPSILON)
142 + /* Use symmetries to perform the calculation in the first quadrant. */ 150 + /* Use symmetries to perform the calculation in the first quadrant. */
143 + __FLT_TYPE arz = __FLT_ABI(fabs) (__real__ z); 151 + arz = __FLT_ABI(fabs) (__real__ z);
144 + __FLT_TYPE aiz = __FLT_ABI(fabs) (__imag__ z); 152 + aiz = __FLT_ABI(fabs) (__imag__ z);
145 + 153 +
146 + if (arz >= __FLT_CST(1.0)/__FLT_EPSILON 154 + if (arz >= __FLT_CST(1.0)/__FLT_EPSILON
147 + || aiz >= __FLT_CST(1.0)/__FLT_EPSILON) 155 + || aiz >= __FLT_CST(1.0)/__FLT_EPSILON)
148 { 156 {
149 /* For large z, z + sqrt(z*z + 1) is approximately 2*z. 157 /* For large z, z + sqrt(z*z + 1) is approximately 2*z.
192 + return ret; 200 + return ret;
193 } 201 }
194 -- 202 --
195 2.35.3.windows.1 203 2.35.3.windows.1
196 204
197 From 099f1cd2d4261573d584aa415f47742fa977c773 Mon Sep 17 00:00:00 2001 205 From f309692d67674526ec9a87ecae527cb6c207cf93 Mon Sep 17 00:00:00 2001
198 From: =?UTF-8?q?Markus=20M=C3=BCtzel?= <markus.muetzel@gmx.de> 206 From: =?UTF-8?q?Markus=20M=C3=BCtzel?= <markus.muetzel@gmx.de>
199 Date: Sun, 7 Aug 2022 13:23:02 +0200 207 Date: Mon, 8 Aug 2022 18:02:40 +0200
200 Subject: [PATCH 2/2] casinh: Use approximation for small real part and 208 Subject: [PATCH 2/3] casinh: Use approximations for small real part and
201 absolute imaginary part < 1 209 absolute imaginary part < 1
202 210
203 --- 211 ---
204 mingw-w64-crt/complex/casinh.def.h | 31 ++++++++++++++++++++++++++++++ 212 mingw-w64-crt/complex/casinh.def.h | 31 ++++++++++++++++++++++++++++++
205 1 file changed, 31 insertions(+) 213 1 file changed, 31 insertions(+)
206 214
207 From 575a070e3fb360fced10a458e72a298d3071c42e Mon Sep 17 00:00:00 2001
208 From: =?UTF-8?q?Markus=20M=C3=BCtzel?= <markus.muetzel@gmx.de>
209 Date: Sun, 7 Aug 2022 19:05:21 +0200
210 Subject: [PATCH 2/2] casinh: Use approximations for small real part and
211 absolute imaginary part < 1
212
213 ---
214 mingw-w64-crt/complex/casinh.def.h | 31 ++++++++++++++++++++++++++++++
215 1 file changed, 31 insertions(+)
216
217 diff --git a/mingw-w64-crt/complex/casinh.def.h b/mingw-w64-crt/complex/casinh.def.h 215 diff --git a/mingw-w64-crt/complex/casinh.def.h b/mingw-w64-crt/complex/casinh.def.h
218 index d309e6392..a5447bfcd 100644 216 index 40ff67579..8d61b1399 100644
219 --- a/mingw-w64-crt/complex/casinh.def.h 217 --- a/mingw-w64-crt/complex/casinh.def.h
220 +++ b/mingw-w64-crt/complex/casinh.def.h 218 +++ b/mingw-w64-crt/complex/casinh.def.h
221 @@ -103,6 +103,37 @@ __FLT_ABI(casinh) (__FLT_TYPE __complex__ z) 219 @@ -104,6 +104,37 @@ __FLT_ABI(casinh) (__FLT_TYPE __complex__ z)
222 ret = __FLT_ABI(clog) (x); 220 ret = __FLT_ABI(clog) (x);
223 __real__ ret += M_LN2; 221 __real__ ret += M_LN2;
224 } 222 }
223 + else if (arz <= __FLT_EPSILON && aiz < __FLT_CST(1.0))
224 + {
225 + /* Taylor series expansion around arz=0 for z + sqrt(z*z + 1):
226 + c = arz + sqrt(1-aiz^2) + i*(aiz + arz*aiz / sqrt(1-aiz^2)) + O(arz^2)
227 + Identity: clog(c) = log(|c|) + i*arg(c)
228 + For real part of result:
229 + |c| = 1 + arz / sqrt(1-aiz^2) + O(arz^2) (Taylor series expansion)
230 + For imaginary part of result:
231 + c = (arz + sqrt(1-aiz^2))/sqrt(1-aiz^2) * (sqrt(1-aiz^2) + i*aiz) + O(arz^6)
232 + */
233 + __FLT_TYPE s1maiz2 = __FLT_ABI(sqrt) ((__FLT_CST(1.0)+aiz)*(__FLT_CST(1.0)-aiz));
234 + __real__ ret = __FLT_ABI(log1p) (arz / s1maiz2);
235 + __imag__ ret = __FLT_ABI(atan2) (aiz, s1maiz2);
236 + }
225 + else if (arz*arz <= __FLT_EPSILON && aiz < __FLT_CST(1.0)) 237 + else if (arz*arz <= __FLT_EPSILON && aiz < __FLT_CST(1.0))
226 + { 238 + {
227 + /* Taylor series expansion around arz=0 for z + sqrt(z*z + 1): 239 + /* Taylor series expansion around arz=0 for z + sqrt(z*z + 1):
228 + c = arz + sqrt(1-aiz^2) + arz^2 / (2*(1-aiz^2)^(3/2)) + i*(aiz + arz*aiz / sqrt(1-aiz^2)) + O(arz^4) 240 + c = arz + sqrt(1-aiz^2) + arz^2 / (2*(1-aiz^2)^(3/2)) + i*(aiz + arz*aiz / sqrt(1-aiz^2)) + O(arz^4)
229 + Identity: clog(c) = log(|c|) + i*arg(c) 241 + Identity: clog(c) = log(|c|) + i*arg(c)
237 + __FLT_TYPE arz2red = arz * arz / __FLT_CST(2.0) / s1maiz2; 249 + __FLT_TYPE arz2red = arz * arz / __FLT_CST(2.0) / s1maiz2;
238 + __real__ ret = __FLT_ABI(log1p) ((arz + arz2red) / s1maiz2); 250 + __real__ ret = __FLT_ABI(log1p) ((arz + arz2red) / s1maiz2);
239 + __imag__ ret = __FLT_ABI(atan2) (aiz * (s1maiz2 + arz), 251 + __imag__ ret = __FLT_ABI(atan2) (aiz * (s1maiz2 + arz),
240 + onemaiz2 + arz*s1maiz2 + arz2red); 252 + onemaiz2 + arz*s1maiz2 + arz2red);
241 + } 253 + }
242 + else if (arz <= __FLT_EPSILON && aiz < __FLT_CST(1.0))
243 + {
244 + /* Taylor series expansion around arz=0 for z + sqrt(z*z + 1):
245 + c = arz + sqrt(1-aiz^2) + i*(aiz + arz*aiz / sqrt(1-aiz^2)) + O(arz^2)
246 + Identity: clog(c) = log(|c|) + i*arg(c)
247 + For real part of result:
248 + |c| = 1 + arz / sqrt(1-aiz^2) + O(arz^2) (Taylor series expansion)
249 + For imaginary part of result:
250 + c = (arz + sqrt(1-aiz^2))/sqrt(1-aiz^2) * (sqrt(1-aiz^2) + i*aiz) + O(arz^6)
251 + */
252 + __FLT_TYPE s1maiz2 = __FLT_ABI(sqrt) ((__FLT_CST(1.0)+aiz)*(__FLT_CST(1.0)-aiz));
253 + __real__ ret = __FLT_ABI(log1p) (arz / s1maiz2);
254 + __imag__ ret = __FLT_ABI(atan2) (aiz, s1maiz2);
255 + }
256 else 254 else
257 { 255 {
258 __real__ x = (arz - aiz) * (arz + aiz) + __FLT_CST(1.0); 256 __real__ x = (arz - aiz) * (arz + aiz) + __FLT_CST(1.0);
259 -- 257 --
260 2.35.3.windows.1 258 2.35.3.windows.1
261 259
260 From b2ad7c96b759e8bc3af504018b70c91c8b0f135c Mon Sep 17 00:00:00 2001
261 From: =?UTF-8?q?Markus=20M=C3=BCtzel?= <markus.muetzel@gmx.de>
262 Date: Mon, 8 Aug 2022 18:03:49 +0200
263 Subject: [PATCH 3/3] catanh: Use approximations for small real part
264
265 ---
266 mingw-w64-crt/complex/catanh.def.h | 37 +++++++++++++++++++++++++-----
267 1 file changed, 31 insertions(+), 6 deletions(-)
268
269 From 7589dfe1f5478a3ba7b93ea6dc28fc0ffd271f12 Mon Sep 17 00:00:00 2001
270 From: =?UTF-8?q?Markus=20M=C3=BCtzel?= <markus.muetzel@gmx.de>
271 Date: Mon, 8 Aug 2022 18:26:20 +0200
272 Subject: [PATCH 3/3] catanh: Use approximations for small real part
273
274 ---
275 mingw-w64-crt/complex/catanh.def.h | 37 +++++++++++++++++++++++++-----
276 1 file changed, 31 insertions(+), 6 deletions(-)
277
278 diff --git a/mingw-w64-crt/complex/catanh.def.h b/mingw-w64-crt/complex/catanh.def.h
279 index 68949d139..1d46d4dcd 100644
280 --- a/mingw-w64-crt/complex/catanh.def.h
281 +++ b/mingw-w64-crt/complex/catanh.def.h
282 @@ -75,17 +75,42 @@ __FLT_ABI(catanh) (__FLT_TYPE __complex__ z)
283 if (r_class == FP_ZERO && i_class == FP_ZERO)
284 return z;
285
286 + /* catanh(z) = 1/2 * clog(1+z) - 1/2 * clog(1-z) = 1/2 * clog((1+z)/(1-z)) */
287 +
288 + /* Use identity clog(c) = 1/2*log(|c|^2) + i*arg(c) to calculate real and
289 + imaginary parts separately. */
290 +
291 + /* real part */
292 + /* |c|^2 = (Im(z)^2 + (1+Re(z))^2)/(Im(z)^2 + (1-Re(z))^2) */
293 i2 = __imag__ z * __imag__ z;
294
295 - n = __FLT_CST(1.0) + __real__ z;
296 - n = i2 + n * n;
297 + if (__FLT_ABI(fabs) (__real__ z) <= __FLT_EPSILON)
298 + {
299 + /* |c|^2 = 1 + 4*Re(z)/(1+Im(z)^2) + O(Re(z)^2) (Taylor series) */
300 + __real__ ret = __FLT_CST(0.25) *
301 + __FLT_ABI(log1p) (__FLT_CST(4.0)*(__real__ z) / (__FLT_CST(1.0) + i2));
302 + }
303 + else if ((__real__ z)*(__real__ z) <= __FLT_EPSILON)
304 + {
305 + /* |c|^2 = 1 + 4*Re(z)/(1+Im(z)^2) + 8*Re(z)^2/(1+Im(z)^2)^2 + O(Re(z)^3) (Taylor series) */
306 + d = __real__ z / (__FLT_CST(1.0) + i2);
307 + __real__ ret = __FLT_CST(0.25) *
308 + __FLT_ABI(log1p) (__FLT_CST(4.0) * d * (__FLT_CST(1.0) + __FLT_CST(2.0) * d));
309 + }
310 + else
311 + {
312 + n = __FLT_CST(1.0) + __real__ z;
313 + n = i2 + n * n;
314
315 - d = __FLT_CST(1.0) - __real__ z;
316 - d = i2 + d * d;
317 + d = __FLT_CST(1.0) - __real__ z;
318 + d = i2 + d * d;
319
320 - __real__ ret = __FLT_CST(0.25) * (__FLT_ABI(log) (n) - __FLT_ABI(log) (d));
321 + __real__ ret = __FLT_CST(0.25) * (__FLT_ABI(log) (n) - __FLT_ABI(log) (d));
322 + }
323
324 - d = 1 - __real__ z * __real__ z - i2;
325 + /* imaginary part */
326 + /* z = (1 - Re(z)^2 - Im(z)^2 + 2i * Im(z) / ((1-Re(z))^2 + Im(z)^2) */
327 + d = __FLT_CST(1.0) - __real__ z * __real__ z - i2;
328
329 __imag__ ret = __FLT_CST(0.5) * __FLT_ABI(atan2) (__FLT_CST(2.0) * __imag__ z, d);
330
331 --
332 2.35.3.windows.1
333