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5624e564 | 1 | /* Copyright (C) 2008-2015 Free Software Foundation, Inc. |
d38a64b4 JR |
2 | Contributor: Joern Rennecke <joern.rennecke@embecosm.com> |
3 | on behalf of Synopsys Inc. | |
4 | ||
5 | This file is part of GCC. | |
6 | ||
7 | GCC is free software; you can redistribute it and/or modify it under | |
8 | the terms of the GNU General Public License as published by the Free | |
9 | Software Foundation; either version 3, or (at your option) any later | |
10 | version. | |
11 | ||
12 | GCC is distributed in the hope that it will be useful, but WITHOUT ANY | |
13 | WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
14 | FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
15 | for more details. | |
16 | ||
17 | Under Section 7 of GPL version 3, you are granted additional | |
18 | permissions described in the GCC Runtime Library Exception, version | |
19 | 3.1, as published by the Free Software Foundation. | |
20 | ||
21 | You should have received a copy of the GNU General Public License and | |
22 | a copy of the GCC Runtime Library Exception along with this program; | |
23 | see the files COPYING3 and COPYING.RUNTIME respectively. If not, see | |
24 | <http://www.gnu.org/licenses/>. */ | |
25 | ||
26 | /* We use a polynom similar to a Tchebycheff polynom to get an initial | |
27 | seed, and then use a newton-raphson iteration step to get an | |
28 | approximate result | |
29 | If this result can't be rounded to the exact result with confidence, we | |
30 | round to the value between the two closest representable values, and | |
31 | test if the correctly rounded value is above or below this value. | |
32 | ||
33 | Because of the Newton-raphson iteration step, an error in the seed at X | |
34 | is amplified by X. Therefore, we don't want a Tchebycheff polynom | |
35 | or a polynom that is close to optimal according to the maximum norm | |
36 | on the errro of the seed value; we want one that is close to optimal | |
37 | according to the maximum norm on the error of the result, i.e. we | |
38 | want the maxima of the polynom to increase linearily. | |
39 | Given an interval [X0,X2) over which to approximate, | |
40 | with X1 := (X0+X2)/2, D := X1-X0, F := 1/D, and S := D/X1 we have, | |
41 | like for Tchebycheff polynoms: | |
42 | P(0) := 1 | |
43 | but then we have: | |
44 | P(1) := X + S*D | |
45 | P(2) := 2 * X^2 + S*D * X - D^2 | |
46 | Then again: | |
47 | P(n+1) := 2 * X * P(n) - D^2 * P (n-1) | |
48 | */ | |
49 | ||
50 | int | |
51 | main (void) | |
52 | { | |
53 | long double T[5]; /* Taylor polynom */ | |
54 | long double P[5][5]; | |
55 | int i, j; | |
56 | long double X0, X1, X2, S; | |
57 | long double inc = 1./64; | |
58 | long double D = inc*0.5; | |
59 | long i0, i1, i2; | |
60 | ||
61 | memset (P, 0, sizeof (P)); | |
62 | P[0][0] = 1.; | |
63 | for (i = 1; i < 5; i++) | |
64 | P[i][i] = 1 << i-1; | |
65 | P[2][0] = -D*D; | |
66 | for (X0 = 1.; X0 < 2.; X0 += inc) | |
67 | { | |
68 | X1 = X0 + inc * 0.5; | |
69 | X2 = X1 + inc; | |
70 | S = D / X1; | |
71 | T[0] = 1./X1; | |
72 | for (i = 1; i < 5; i++) | |
73 | T[i] = T[i-1] * -T[0]; | |
74 | #if 0 | |
75 | printf ("T %1.8f %f %f %f %f\n", (double)T[0], (double)T[1], (double)T[2], | |
76 | (double)T[3], (double)T[4]); | |
77 | #endif | |
78 | P[1][0] = S*D; | |
79 | P[2][1] = S*D; | |
80 | for (i = 3; i < 5; i++) | |
81 | { | |
82 | P[i][0] = -D*D*P[i-2][0]; | |
83 | for (j = 1; j < i; j++) | |
84 | P[i][j] = 2*P[i-1][j-1]-D*D*P[i-2][j]; | |
85 | } | |
86 | #if 0 | |
87 | printf ("P3 %1.8f %f %f %f %f\n", (double)P[3][0], (double)P[3][1], (double)P[3][2], | |
88 | (double)P[3][3], (double)P[3][4]); | |
89 | printf ("P4 %1.8f %f %f %f %f\n", (double)P[4][0], (double)P[4][1], (double)P[4][2], | |
90 | (double)P[4][3], (double)P[4][4]); | |
91 | #endif | |
92 | for (i = 4; i > 1; i--) | |
93 | { | |
94 | long double a = T[i]/P[i][i]; | |
95 | ||
96 | for (j = 0; j < i; j++) | |
97 | T[j] -= a * P[i][j]; | |
98 | } | |
99 | #if 0 | |
100 | printf ("A %1.8f %f %f\n", (double)T[0], (double)T[1], (double)T[2]); | |
101 | #endif | |
102 | #if 0 | |
103 | i2 = T[2]*512; | |
104 | long double a = (T[2]-i/512.)/P[2][2]; | |
105 | for (j = 0; j < 2; j++) | |
106 | T[j] -= a * P[2][j]; | |
107 | #else | |
108 | i2 = 0; | |
109 | #endif | |
110 | for (i = 0, i0 = 0; i < 4; i++) | |
111 | { | |
112 | long double T0, Ti1; | |
113 | ||
114 | i1 = T[1]*8192. + i0 / (long double)(1 << 19) - 0.5; | |
115 | i1 = - (-i1 & 0x1fff); | |
116 | Ti1 = ((unsigned)(-i1 << 19) | i0) /-(long double)(1LL<<32LL); | |
117 | T0 = T[0] - (T[1]-Ti1)/P[1][1] * P[1][0] - (X1 - 1) * Ti1; | |
118 | i0 = T0 * 512 * 1024 + 0.5; | |
119 | i0 &= 0x7ffff; | |
120 | } | |
121 | #if 0 | |
122 | printf ("A %1.8f %f %f\n", (double)T[0], (double)T[1], (double)T[2]); | |
123 | #endif | |
124 | printf ("\t.long 0x%x\n", (-i1 << 19) | i0); | |
125 | } | |
126 | return 0; | |
127 | } |