pxfm/logs/
fast_log_dd.rs

1/*
2 * // Copyright (c) Radzivon Bartoshyk 8/2025. All rights reserved.
3 * //
4 * // Redistribution and use in source and binary forms, with or without modification,
5 * // are permitted provided that the following conditions are met:
6 * //
7 * // 1.  Redistributions of source code must retain the above copyright notice, this
8 * // list of conditions and the following disclaimer.
9 * //
10 * // 2.  Redistributions in binary form must reproduce the above copyright notice,
11 * // this list of conditions and the following disclaimer in the documentation
12 * // and/or other materials provided with the distribution.
13 * //
14 * // 3.  Neither the name of the copyright holder nor the names of its
15 * // contributors may be used to endorse or promote products derived from
16 * // this software without specific prior written permission.
17 * //
18 * // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
19 * // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
21 * // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
22 * // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
24 * // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
25 * // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
26 * // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
27 * // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
28 */
29
30/* Given 2^-1074 <= x <= 0x1.fffffffffffffp+1023, this routine puts in h+l
31   an approximation of log(x) such that |l| < 2^-23.89*|h| and
32
33   | h + l - log(x) | <= elog * |log x|
34
35   with elog = 2^-73.527  if x < 1/sqrt(2) or sqrt(2) < x,
36   and  elog = 2^-67.0544 if 1/sqrt(2) < x < sqrt(2)
37   (note that x cannot equal 1/sqrt(2) nor sqrt(2)).
38*/
39use crate::common::dd_fmla;
40use crate::double_double::DoubleDouble;
41use crate::pow_exec::log_poly_1;
42use crate::pow_tables::POW_INVERSE;
43
44// Generated by SageMath:
45// values = POW_INVERSE
46// R = RealField(150)
47//
48// def hex_to_float(h):
49//     return struct.unpack('>d', struct.pack('>Q', h))[0]
50//
51// real_array = [R(hex_to_float(h)) for h in values]
52//
53// for r in real_array:
54//     print_double_double("", -RealField(180)(r).log())
55pub(crate) static FAST_LOG_DD_INV: [(u64, u64); 182] = [
56    (0x3c6bc60efafc6f6e, 0xbfd5ff3070a793d4),
57    (0x3c78ebcb7dee9a3d, 0xbfd5a42ab0f4cfe2),
58    (0x3c6819cf7e308ddb, 0xbfd548a2c3add263),
59    (0x3c742a87d977dc5e, 0xbfd4ec973260026a),
60    (0x3c69ffc341f177dc, 0xbfd49006804009d1),
61    (0x3c729931715ac903, 0xbfd432ef2a04e814),
62    (0x3c70bcfb6082ce6d, 0xbfd404308686a7e4),
63    (0x3c6c68651945f97c, 0xbfd3a64c556945ea),
64    (0x3c64dd4c580919f8, 0xbfd347dd9a987d55),
65    (0x3c78f4cdb95ebdf9, 0xbfd2e8e2bae11d31),
66    (0xbc77ad24c13f040e, 0xbfd2895a13de86a3),
67    (0x3c776f5eb09628af, 0xbfd22941fbcf7966),
68    (0x3c7c9fdf9a0c4b07, 0xbfd1f8ff9e48a2f3),
69    (0xbc79d3d1b0e4d147, 0xbfd1980d2dd4236f),
70    (0xbc77b66298edd24a, 0xbfd136870293a8b0),
71    (0xbc589fa0ab4cb31d, 0xbfd1058bf9ae4ad5),
72    (0xbc77dcfde8061c03, 0xbfd0a324e27390e3),
73    (0x3c628ec217a5022d, 0xbfd0402594b4d041),
74    (0x3c6caaae64f21acb, 0xbfcfb9186d5e3e2b),
75    (0xbc2c5f6dfd018c37, 0xbfcf550a564b7b37),
76    (0x3c46e03a39bfc89b, 0xbfce8c0252aa5a60),
77    (0x3c461578001e0162, 0xbfce27076e2af2e6),
78    (0xbc66e443597e4d40, 0xbfcd5c216b4fbb91),
79    (0x3c64f689f8434012, 0xbfcc8ff7c79a9a22),
80    (0x3c673dee38a3fb6b, 0xbfcc2968558c18c1),
81    (0xbc6ba27fdc19e1a0, 0xbfcb5b519e8fb5a4),
82    (0x3c5398cff3641985, 0xbfcaf3c94e80bff3),
83    (0xbc493711b07a998c, 0xbfca23bc1fe2b563),
84    (0xbc6575e31f003e0c, 0xbfc9bb362e7dfb83),
85    (0x3c6569d851a56770, 0xbfc8e928de886d41),
86    (0x3c6bf7fdbfa08d9a, 0xbfc87fa06520c911),
87    (0xbc4be36b2d6a0608, 0xbfc7ab890210d909),
88    (0x3c5b264062a84cdb, 0xbfc740f8f54037a5),
89    (0x3c6caae268ecd179, 0xbfc6d60fe719d21d),
90    (0x3c5bc60efafc6f6e, 0xbfc5ff3070a793d4),
91    (0x3c565d22aa8ad7cf, 0xbfc59338d9982086),
92    (0xbc668981bcc36756, 0xbfc4ba36f39a55e5),
93    (0xbc69f4f6543e1f88, 0xbfc44d2b6ccb7d1e),
94    (0x3c5ab3a8e7d81017, 0xbfc3dfc2b0ecc62a),
95    (0x3c06b9c7d96091fa, 0xbfc303d718e47fd3),
96    (0xbc6301771c407dbf, 0xbfc29552f81ff523),
97    (0xbc6f547bf1809e88, 0xbfc2266f190a5acb),
98    (0xbc6a28813e3a7f07, 0xbfc14785846742ac),
99    (0xbc69a5dc5e9030ac, 0xbfc0d77e7cd08e59),
100    (0xbc550c647eb86499, 0xbfc0671512ca596e),
101    (0xbc585f325c5bbacd, 0xbfbf0a30c01162a6),
102    (0x3c361578001e0162, 0xbfbe27076e2af2e6),
103    (0xbc5790dd951d90fa, 0xbfbd4313d66cb35d),
104    (0xbc35d617ef8161b1, 0xbfbc5e548f5bc743),
105    (0xbc5942f48aa70ea9, 0xbfba926d3a4ad563),
106    (0x3c42099e1c184e8e, 0xbfb9ab42462033ad),
107    (0x3c24a697ab3424a9, 0xbfb8c345d6319b21),
108    (0x3c5eeedfcdd94131, 0xbfb7da766d7b12cd),
109    (0x3c5388458ec21b6a, 0xbfb60658a93750c4),
110    (0xbc5a49e39a1a8be4, 0xbfb51b073f06183f),
111    (0xbc4ddd4f935996c9, 0xbfb42edcbea646f0),
112    (0x3c5b599f227becbb, 0xbfb341d7961bd1d1),
113    (0x3c1c125963fc4cfd, 0xbfb253f62f0a1417),
114    (0x3c379da3e8c22cda, 0xbfb16536eea37ae1),
115    (0xbc485f325c5bbacd, 0xbfaf0a30c01162a6),
116    (0xbc21e3c53257fd47, 0xbfad276b8adb0b52),
117    (0x3c3eb9759c130499, 0xbfab42dd711971bf),
118    (0xbc4f5a0e80520bf2, 0xbfa95c830ec8e3eb),
119    (0xbc418d3ca87b9296, 0xbfa77458f632dcfc),
120    (0x3c4ce55c2b4e2b72, 0xbfa58a5bafc8e4d5),
121    (0x3c45bfa937f551bb, 0xbfa39e87b9febd60),
122    (0x3c3e9ae889bac481, 0xbfa1b0d98923d980),
123    (0xbc333e3f04f1ef23, 0xbf9f829b0e783300),
124    (0x3bf0ae69229dc868, 0xbf9b9fc027af9198),
125    (0x3c35b602ace3a510, 0xbf97b91b07d5b11b),
126    (0x3c10cb5a902b3a1c, 0xbf93cea44346a575),
127    (0x3c183092c59642a1, 0xbf8fc0a8b0fc03e4),
128    (0x3c116d7687d3df21, 0xbf87dc475f810a77),
129    (0x3bce44b7e3711ebf, 0xbf7fe02a6b106789),
130    (0x0000000000000000, 0x0000000000000000),
131    (0x0000000000000000, 0x0000000000000000),
132    (0x3bec14b9f9377a1d, 0x3f78121214586b54),
133    (0xbc2c5517f64bc223, 0x3f841929f96832f0),
134    (0x3c2806208c04c220, 0x3f8c317384c75f06),
135    (0xbc2cd7b66e01c26d, 0x3f9228fb1fea2e28),
136    (0xbbf8ed4d357c9c97, 0x3f963d6178690bd6),
137    (0x3c1ec1a5f86d41f9, 0x3f9a55f548c5c43f),
138    (0x3c375b44595cab18, 0x3f9e72bf2813ce51),
139    (0x3c4c05cf1d753622, 0x3fa0415d89e74444),
140    (0xbc4947f792615916, 0x3fa252f32f8d183f),
141    (0xbc4cdd6f7f4a137e, 0x3fa466aed42de3ea),
142    (0x3c40413e6505e603, 0x3fa67c94f2d4bb58),
143    (0x3c3a8be97660a23d, 0x3fa894aa149fb343),
144    (0x3c2a353bb42e0add, 0x3faaaef2d0fb10fc),
145    (0x3c3e5cf3a0f56f72, 0x3fabbcebfc68f420),
146    (0x3c44e6c986f44c55, 0x3fadda8adc67ee4e),
147    (0xbc4cd9f1f95c2eed, 0x3faffa6911ab9301),
148    (0xbc5a4a128d192686, 0x3fb10e45b3cae831),
149    (0xbc5cc0fbce104eaa, 0x3fb2207b5c78549e),
150    (0xbc5d15d38d2fa3f7, 0x3fb2aa04a44717a5),
151    (0x3c47a976d3b5b45f, 0x3fb3bdf5a7d1ee64),
152    (0x3c5769f42c7842cc, 0x3fb4d3115d207eac),
153    (0xbc545f9d61c68c1b, 0x3fb55e10050e0384),
154    (0xbc59acd8b33f8fdc, 0x3fb674f089365a7a),
155    (0x3c5abca5b4fdb880, 0x3fb78d02263d82d3),
156    (0x3c3b9f2dffbeed43, 0x3fb8197e2f40e3f0),
157    (0xbc5478a85704ccb7, 0x3fb9335e5d594989),
158    (0xbc55b5ca203e4259, 0x3fba4e7640b1bc38),
159    (0x3c537d8f39bee659, 0x3fbadc77ee5aea8c),
160    (0xbc4cdc9f6f5f38c7, 0x3fbbf968769fca11),
161    (0x3c49daf7df76ad2a, 0x3fbd179788219364),
162    (0x3c5401fa71733019, 0x3fbda727638446a2),
163    (0xbc4a2bf991780d3f, 0x3fbec739830a1120),
164    (0xbc59361574fb24e2, 0x3fbf57bc7d9005db),
165    (0x3c639e2d3f8b7d10, 0x3fc03cdc0a51ec0d),
166    (0xbc6dd7009902bf32, 0x3fc08598b59e3a07),
167    (0xbc50e63a5f01c691, 0x3fc1178e8227e47c),
168    (0xbc62d56ff61c2bfb, 0x3fc160c8024b27b1),
169    (0x3c462c9ef939ac5d, 0x3fc1f3b925f25d41),
170    (0xbc66e38161051d69, 0x3fc23d712a49c202),
171    (0xbc5499a3f25af95f, 0x3fc2d1610c86813a),
172    (0xbc5c4716bdfc0cc9, 0x3fc31b994d3a4f85),
173    (0x3c370d6cdf05266c, 0x3fc3b08b6757f2a9),
174    (0xbc6d87e6a354d056, 0x3fc3fb45a59928cc),
175    (0xbc50d5604930f135, 0x3fc4913d8333b561),
176    (0xbc6927d47803c5f4, 0x3fc4dc7b897bc1c8),
177    (0x3c64f4d710fec38e, 0x3fc5737cc9018cdd),
178    (0xbc21f5b44c0df7e7, 0x3fc5bf406b543db2),
179    (0xbc3d34f0f4621bed, 0x3fc6574ebe8c133a),
180    (0x3c696332bd4b341f, 0x3fc6a399dabbd383),
181    (0xbc68de59c21e166c, 0x3fc6f0128b756abc),
182    (0x3c5ef8f6ebcfb201, 0x3fc7898d85444c73),
183    (0xbc4ac5f0c075b847, 0x3fc7d6903caf5ad0),
184    (0x3c6d685f35eea2a0, 0x3fc871213750e994),
185    (0x3c555aa8b6997a40, 0x3fc8beafeb38fe8c),
186    (0x3c6054473941ad99, 0x3fc90c6db9fcbcd9),
187    (0x3c6f47dfd871f87f, 0x3fc9a8778debaa38),
188    (0x3c435a19605e67ef, 0x3fc9f6c407089664),
189    (0x3c5df207dc5c34c6, 0x3fca454082e6ab05),
190    (0x3c6ab5ca9eaa088a, 0x3fcae2ca6f672bd4),
191    (0xbc66353ab386a94d, 0x3fcb31d8575bce3d),
192    (0x3c3a0ee735d9f0ec, 0x3fcb811730b823d2),
193    (0x3c3dd355f6a516d7, 0x3fcbd087383bd8ad),
194    (0xbc68e58b2c57a4a5, 0x3fcc6ffbc6f00f71),
195    (0x3c653d154280394f, 0x3fccc000c9db3c52),
196    (0x3c660629242471a2, 0x3fcd1037f2655e7b),
197    (0x3c5aa11d49f96cb9, 0x3fcdb13db0d48940),
198    (0x3c5fea48dd7b81d1, 0x3fce020cc6235ab5),
199    (0x3c42276041f43042, 0x3fce530effe71012),
200    (0xbc6d33919ab94074, 0x3fcea4449f04aaf5),
201    (0xbc527c77ded76aad, 0x3fcf474b134df229),
202    (0x3c6f665066f980a2, 0x3fcf991c6cb3b379),
203    (0x3c28de00938b4c40, 0x3fcfeb2233ea07cd),
204    (0xbc418290bd2932e2, 0x3fd01eae5626c691),
205    (0xbc70779634061cbc, 0x3fd047e60cde83b8),
206    (0x3c643c2e68684d53, 0x3fd09aa572e6c6d4),
207    (0x3c5162c79d5d11ee, 0x3fd0c42d676162e3),
208    (0xbc692b49ef282b09, 0x3fd0edd060b78081),
209    (0xbc60e63a5f01c691, 0x3fd1178e8227e47c),
210    (0x3c1e0936abd4fa6e, 0x3fd14167ef367783),
211    (0x3c766fbd28b40935, 0x3fd16b5ccbacfb73),
212    (0xbc612aeb84249223, 0x3fd1bf99635a6b95),
213    (0x3c7512c3749a1e4e, 0x3fd1e9e1678899f4),
214    (0x3c6f7ae91aeba60a, 0x3fd214456d0eb8d4),
215    (0x3c3bb75d1addf870, 0x3fd23ec5991eba49),
216    (0x3c7e0efadd9db02b, 0x3fd269621134db92),
217    (0xbc6856e61c515740, 0x3fd2941afb186b7c),
218    (0xbc782dad7fd86088, 0x3fd2bef07cdc9354),
219    (0xbc73d69909e5c3dc, 0x3fd314f1e1d35ce4),
220    (0xbc5cd55b8a4746c0, 0x3fd3401e12aecba1),
221    (0xbc5324f0e883858e, 0x3fd36b6776be1117),
222    (0xbc5ce2b31b31e8b0, 0x3fd396ce359bbf54),
223    (0xbc72ad27e50a8ec6, 0x3fd3c25277333184),
224    (0x3c783d680d3c1084, 0x3fd3edf463c1683e),
225    (0x3c60dbb243827392, 0x3fd419b423d5e8c7),
226    (0xbc72b125247b0fa5, 0x3fd44591e0539f49),
227    (0x3c38fb4c14c56eef, 0x3fd4718dc271c41b),
228    (0xbc69964a168ccaca, 0x3fd49da7f3bcc41f),
229    (0xbc5123615b147a5d, 0x3fd4c9e09e172c3c),
230    (0xbc758cb3124b9245, 0x3fd4f637ebba9810),
231    (0xbc68f7e9b38a6979, 0x3fd522ae0738a3d8),
232    (0xbc7aacfdbbdab914, 0x3fd54f431b7be1a9),
233    (0xbc60908d15f88b63, 0x3fd57bf753c8d1fb),
234    (0xbc5e6c2bdfb3e037, 0x3fd5a8cadbbedfa1),
235    (0xbc76541148cbb8a2, 0x3fd5d5bddf595f30),
236    (0xbc56e8920c09b73f, 0x3fd602d08af091ec),
237    (0x3c6dc18ce51fff99, 0x3fd630030b3aac49),
238];
239
240#[inline]
241pub(crate) fn fast_log_dd(ddx: DoubleDouble) -> DoubleDouble {
242    // We'll compute log((z+1)+1) as log(xh+xl) = log(xh) + log(1+xl/xh).
243    // since xl/xh < ulp(xh) we'll use for log(1+xl/xh)
244    // one taylor term what means that log(1+xl/xh) = log_lo + O(x^2)
245    let log_lo = if ddx.hi <= f64::from_bits(0x7fd0000000000000) || ddx.lo.abs() >= 4.0 {
246        ddx.lo / ddx.hi
247    } else {
248        0.
249    }; // avoid spurious underflow
250
251    let x_u = ddx.hi.to_bits();
252    let mut m = x_u & 0xfffffffffffff;
253    let mut e: i64 = ((x_u >> 52) & 0x7ff) as i64;
254
255    let t;
256    if e != 0 {
257        t = m | (0x3ffu64 << 52);
258        m = m.wrapping_add(1u64 << 52);
259        e -= 0x3ff;
260    } else {
261        /* x is a subnormal double  */
262        let k = m.leading_zeros() - 11;
263
264        e = -0x3fei64 - k as i64;
265        m = m.wrapping_shl(k);
266        t = m | (0x3ffu64 << 52);
267    }
268
269    /* now |x| = 2^_e*_t = 2^(_e-52)*m with 1 <= _t < 2,
270    and 2^52 <= _m < 2^53 */
271
272    //   log(x) = log(t) + E · log(2)
273    let mut t = f64::from_bits(t);
274
275    // If m > sqrt(2) we divide it by 2 so ensure 1/sqrt(2) < t < sqrt(2)
276    let c: usize = (m >= 0x16a09e667f3bcd) as usize;
277    static CY: [f64; 2] = [1.0, 0.5];
278    static CM: [u64; 2] = [44, 45];
279
280    e = e.wrapping_add(c as i64);
281    let be = e;
282    let i = m >> CM[c]; /* i/2^8 <= t < (i+1)/2^8 */
283    /* when c=1, we have 0x16a09e667f3bcd <= m < 2^53, thus 90 <= i <= 127;
284    when c=0, we have 2^52 <= m < 0x16a09e667f3bcd, thus 128 <= i <= 181 */
285    t *= CY[c];
286    /* now 0x1.6a09e667f3bcdp-1 <= t < 0x1.6a09e667f3bcdp+0,
287    and log(x) = E * log(2) + log(t) */
288
289    let r = f64::from_bits(POW_INVERSE[(i - 181) as usize]);
290    let log_r = DoubleDouble::from_bit_pair(FAST_LOG_DD_INV[(i - 181) as usize]);
291
292    let z = dd_fmla(r, t, -1.0);
293
294    const LOG2_DD: DoubleDouble = DoubleDouble::new(
295        f64::from_bits(0x3c7abc9e3b39803f),
296        f64::from_bits(0x3fe62e42fefa39ef),
297    );
298
299    let dt = DoubleDouble::mul_f64_add(LOG2_DD, be as f64, log_r);
300
301    let mut v = DoubleDouble::f64_add(dt.hi, DoubleDouble::new(dt.lo, z));
302    let p = log_poly_1(z);
303    v = DoubleDouble::f64_add(v.hi, DoubleDouble::new(v.lo + p.lo + log_lo, p.hi));
304
305    v
306}
307
308#[inline]
309pub(crate) fn fast_log_d_to_dd(ddx: f64) -> DoubleDouble {
310    // We'll compute log((z+1)+1) as log(xh+xl) = log(xh) + log(1+xl/xh).
311    // since xl/xh < ulp(xh) we'll use for log(1+xl/xh)
312    // one taylor term what means that log(1+xl/xh) = log_lo + O(x^2)
313
314    let x_u = ddx.to_bits();
315    let mut m = x_u & 0xfffffffffffff;
316    let mut e: i64 = ((x_u >> 52) & 0x7ff) as i64;
317
318    let t;
319    if e != 0 {
320        t = m | (0x3ffu64 << 52);
321        m = m.wrapping_add(1u64 << 52);
322        e -= 0x3ff;
323    } else {
324        /* x is a subnormal double  */
325        let k = m.leading_zeros() - 11;
326
327        e = -0x3fei64 - k as i64;
328        m = m.wrapping_shl(k);
329        t = m | (0x3ffu64 << 52);
330    }
331
332    /* now |x| = 2^_e*_t = 2^(_e-52)*m with 1 <= _t < 2,
333    and 2^52 <= _m < 2^53 */
334
335    //   log(x) = log(t) + E · log(2)
336    let mut t = f64::from_bits(t);
337
338    // If m > sqrt(2) we divide it by 2 so ensure 1/sqrt(2) < t < sqrt(2)
339    let c: usize = (m >= 0x16a09e667f3bcd) as usize;
340    static CY: [f64; 2] = [1.0, 0.5];
341    static CM: [u64; 2] = [44, 45];
342
343    e = e.wrapping_add(c as i64);
344    let be = e;
345    let i = m >> CM[c]; /* i/2^8 <= t < (i+1)/2^8 */
346    /* when c=1, we have 0x16a09e667f3bcd <= m < 2^53, thus 90 <= i <= 127;
347    when c=0, we have 2^52 <= m < 0x16a09e667f3bcd, thus 128 <= i <= 181 */
348    t *= CY[c];
349    /* now 0x1.6a09e667f3bcdp-1 <= t < 0x1.6a09e667f3bcdp+0,
350    and log(x) = E * log(2) + log(t) */
351
352    let r = f64::from_bits(POW_INVERSE[(i - 181) as usize]);
353    let log_r = DoubleDouble::from_bit_pair(FAST_LOG_DD_INV[(i - 181) as usize]);
354
355    let z = dd_fmla(r, t, -1.0);
356
357    const LOG2_DD: DoubleDouble = DoubleDouble::new(
358        f64::from_bits(0x3c7abc9e3b39803f),
359        f64::from_bits(0x3fe62e42fefa39ef),
360    );
361
362    let dt = DoubleDouble::mul_f64_add(LOG2_DD, be as f64, log_r);
363
364    let mut v = DoubleDouble::f64_add(dt.hi, DoubleDouble::new(dt.lo, z));
365    let p = log_poly_1(z);
366    v = DoubleDouble::f64_add(v.hi, DoubleDouble::new(v.lo + p.lo, p.hi));
367
368    DoubleDouble::from_exact_add(v.hi, v.lo)
369}
pxfm/logs/fast_log_dd.rs

pxfm/logs/
fast_log_dd.rs
