pxfm 0.1.29

Fast and accurate math
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163

/*
 * // Copyright (c) Radzivon Bartoshyk 6/2025. All rights reserved.
 * //
 * // Redistribution and use in source and binary forms, with or without modification,
 * // are permitted provided that the following conditions are met:
 * //
 * // 1.  Redistributions of source code must retain the above copyright notice, this
 * // list of conditions and the following disclaimer.
 * //
 * // 2.  Redistributions in binary form must reproduce the above copyright notice,
 * // this list of conditions and the following disclaimer in the documentation
 * // and/or other materials provided with the distribution.
 * //
 * // 3.  Neither the name of the copyright holder nor the names of its
 * // contributors may be used to endorse or promote products derived from
 * // this software without specific prior written permission.
 * //
 * // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 * // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
 * // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
 * // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
 * // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
 * // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
 * // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
 * // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */
use crate::dyadic_float::{DyadicFloat128, DyadicSign};

// Logarithm range reduction - Step 3:
//   r(k) = 2^-21 round(2^21 / (1 + k*2^-21)) for k = -80 .. 80.
// Output range:
//   [-0x1.01928p-22 , 0x1p-22]
// We store S[i] = 2^21 (r(i - 80) - 1).
static S3: [i32; 161] = [
    0x50, 0x4f, 0x4e, 0x4d, 0x4c, 0x4b, 0x4a, 0x49, 0x48, 0x47, 0x46, 0x45, 0x44, 0x43, 0x42, 0x41,
    0x40, 0x3f, 0x3e, 0x3d, 0x3c, 0x3b, 0x3a, 0x39, 0x38, 0x37, 0x36, 0x35, 0x34, 0x33, 0x32, 0x31,
    0x30, 0x2f, 0x2e, 0x2d, 0x2c, 0x2b, 0x2a, 0x29, 0x28, 0x27, 0x26, 0x25, 0x24, 0x23, 0x22, 0x21,
    0x20, 0x1f, 0x1e, 0x1d, 0x1c, 0x1b, 0x1a, 0x19, 0x18, 0x17, 0x16, 0x15, 0x14, 0x13, 0x12, 0x11,
    0x10, 0xf, 0xe, 0xd, 0xc, 0xb, 0xa, 0x9, 0x8, 0x7, 0x6, 0x5, 0x4, 0x3, 0x2, 0x1, 0x0, -0x1,
    -0x2, -0x3, -0x4, -0x5, -0x6, -0x7, -0x8, -0x9, -0xa, -0xb, -0xc, -0xd, -0xe, -0xf, -0x10,
    -0x11, -0x12, -0x13, -0x14, -0x15, -0x16, -0x17, -0x18, -0x19, -0x1a, -0x1b, -0x1c, -0x1d,
    -0x1e, -0x1f, -0x20, -0x21, -0x22, -0x23, -0x24, -0x25, -0x26, -0x27, -0x28, -0x29, -0x2a,
    -0x2b, -0x2c, -0x2d, -0x2e, -0x2f, -0x30, -0x31, -0x32, -0x33, -0x34, -0x35, -0x36, -0x37,
    -0x38, -0x39, -0x3a, -0x3b, -0x3c, -0x3d, -0x3e, -0x3f, -0x40, -0x41, -0x42, -0x43, -0x44,
    -0x45, -0x46, -0x47, -0x48, -0x49, -0x4a, -0x4b, -0x4c, -0x4d, -0x4e, -0x4f, -0x50,
];

// Logarithm range reduction - Step 4
//   r(k) = 2^-28 round(2^28 / (1 + k*2^-28)) for k = -65 .. 64.
// Output range:
//   [-0x1.0002143p-29 , 0x1p-29]
// We store S[i] = 2^28 (r(i - 65) - 1).
static S4: [i32; 130] = [
    0x41, 0x40, 0x3f, 0x3e, 0x3d, 0x3c, 0x3b, 0x3a, 0x39, 0x38, 0x37, 0x36, 0x35, 0x34, 0x33, 0x32,
    0x31, 0x30, 0x2f, 0x2e, 0x2d, 0x2c, 0x2b, 0x2a, 0x29, 0x28, 0x27, 0x26, 0x25, 0x24, 0x23, 0x22,
    0x21, 0x20, 0x1f, 0x1e, 0x1d, 0x1c, 0x1b, 0x1a, 0x19, 0x18, 0x17, 0x16, 0x15, 0x14, 0x13, 0x12,
    0x11, 0x10, 0xf, 0xe, 0xd, 0xc, 0xb, 0xa, 0x9, 0x8, 0x7, 0x6, 0x5, 0x4, 0x3, 0x2, 0x1, 0x0,
    -0x1, -0x2, -0x3, -0x4, -0x5, -0x6, -0x7, -0x8, -0x9, -0xa, -0xb, -0xc, -0xd, -0xe, -0xf,
    -0x10, -0x11, -0x12, -0x13, -0x14, -0x15, -0x16, -0x17, -0x18, -0x19, -0x1a, -0x1b, -0x1c,
    -0x1d, -0x1e, -0x1f, -0x20, -0x21, -0x22, -0x23, -0x24, -0x25, -0x26, -0x27, -0x28, -0x29,
    -0x2a, -0x2b, -0x2c, -0x2d, -0x2e, -0x2f, -0x30, -0x31, -0x32, -0x33, -0x34, -0x35, -0x36,
    -0x37, -0x38, -0x39, -0x3a, -0x3b, -0x3c, -0x3d, -0x3e, -0x3f, -0x40,
];

// Logarithm range reduction - Step 2:
//   r(k) = 2^-16 round(2^16 / (1 + k*2^-14)) for k = -2^6 .. 2^7.
// Output range:
//   [-0x1.3ffcp-15, 0x1.3e3dp-15]
// We store S2[i] = 2^16 (r(i - 2^6) - 1).
static S2: [i32; 193] = [
    0x101, 0xfd, 0xf9, 0xf5, 0xf1, 0xed, 0xe9, 0xe5, 0xe1, 0xdd, 0xd9, 0xd5, 0xd1, 0xcd, 0xc9,
    0xc5, 0xc1, 0xbd, 0xb9, 0xb4, 0xb0, 0xac, 0xa8, 0xa4, 0xa0, 0x9c, 0x98, 0x94, 0x90, 0x8c, 0x88,
    0x84, 0x80, 0x7c, 0x78, 0x74, 0x70, 0x6c, 0x68, 0x64, 0x60, 0x5c, 0x58, 0x54, 0x50, 0x4c, 0x48,
    0x44, 0x40, 0x3c, 0x38, 0x34, 0x30, 0x2c, 0x28, 0x24, 0x20, 0x1c, 0x18, 0x14, 0x10, 0xc, 0x8,
    0x4, 0x0, -0x4, -0x8, -0xc, -0x10, -0x14, -0x18, -0x1c, -0x20, -0x24, -0x28, -0x2c, -0x30,
    -0x34, -0x38, -0x3c, -0x40, -0x44, -0x48, -0x4c, -0x50, -0x54, -0x58, -0x5c, -0x60, -0x64,
    -0x68, -0x6c, -0x70, -0x74, -0x78, -0x7c, -0x80, -0x84, -0x88, -0x8c, -0x90, -0x94, -0x98,
    -0x9c, -0xa0, -0xa4, -0xa8, -0xac, -0xb0, -0xb4, -0xb7, -0xbb, -0xbf, -0xc3, -0xc7, -0xcb,
    -0xcf, -0xd3, -0xd7, -0xdb, -0xdf, -0xe3, -0xe7, -0xeb, -0xef, -0xf3, -0xf7, -0xfb, -0xff,
    -0x103, -0x107, -0x10b, -0x10f, -0x113, -0x117, -0x11b, -0x11f, -0x123, -0x127, -0x12b, -0x12f,
    -0x133, -0x137, -0x13a, -0x13e, -0x142, -0x146, -0x14a, -0x14e, -0x152, -0x156, -0x15a, -0x15e,
    -0x162, -0x166, -0x16a, -0x16e, -0x172, -0x176, -0x17a, -0x17e, -0x182, -0x186, -0x18a, -0x18e,
    -0x192, -0x195, -0x199, -0x19d, -0x1a1, -0x1a5, -0x1a9, -0x1ad, -0x1b1, -0x1b5, -0x1b9, -0x1bd,
    -0x1c1, -0x1c5, -0x1c9, -0x1cd, -0x1d1, -0x1d5, -0x1d9, -0x1dd, -0x1e0, -0x1e4, -0x1e8, -0x1ec,
    -0x1f0, -0x1f4, -0x1f8, -0x1fc,
];

// Perform logarithm range reduction steps 2-4.
// Inputs from the first step of range reduction:
//   m_x : the reduced argument after the first step of range reduction
//         satisfying  -2^-8 <= m_x < 2^-7  and  ulp(m_x) >= 2^-60.
//   idx1: index of the -log(r1) table from the first step.
// Outputs of the extra range reduction steps:
//   sum: adding -log(r1) - log(r2) - log(r3) - log(r4) to the resulted sum.
//   return value: the reduced argument v satisfying:
//                 -0x1.0002143p-29 <= v < 0x1p-29,  and  ulp(v) >= 2^(-125).
pub(crate) fn log_range_reduction(
    m_x: f64,
    log_table: &[&[DyadicFloat128]; 4],
    sum: DyadicFloat128,
) -> (DyadicFloat128, DyadicFloat128) {
    let v = (m_x * f64::from_bits(0x43b0000000000000)) as i64; // ulp = 2^-60

    // Range reduction - Step 2
    // Output range: vv2 in [-0x1.3ffcp-15, 0x1.3e3dp-15].
    // idx2 = trunc(2^14 * (v + 2^-8 + 2^-15))
    let idx2 = ((v.wrapping_add(0x10_2000_0000_0000)) >> 46) as usize;
    let z0 = log_table[1][idx2];
    let mut sum = sum + z0;

    let s2: i64 = S2[idx2] as i64; // |s| <= 2^-7, ulp = 2^-16
    let sv2 = s2 * v; // |s*v| < 2^-14, ulp = 2^(-60-16) = 2^-76
    let spv2 = s2.wrapping_shl(44).wrapping_add(v); // |s + v| < 2^-14, ulp = 2^-60
    let vv2 = spv2.wrapping_shl(16).wrapping_add(sv2); // |vv2| < 2^-14, ulp = 2^-76

    // Range reduction - Step 3
    // Output range: vv3 in [-0x1.01928p-22 , 0x1p-22]
    // idx3 = trunc(2^21 * (v + 80*2^-21 + 2^-22))
    let idx3 = vv2.wrapping_add(0x2840_0000_0000_0000).wrapping_shr(55) as usize;
    sum = sum + log_table[2][idx3];

    let s3 = S3[idx3] as i64; // |s| < 2^-13, ulp = 2^-21
    let spv3: i64 = s3.wrapping_shl(55).wrapping_add(vv2); // |s + v| < 2^-21, ulp = 2^-76
    // |s*v| < 2^-27, ulp = 2^(-76-21) = 2^-97
    let sv3: i128 = s3 as i128 * vv2 as i128;
    // |vv3| < 2^-21, ulp = 2^-97
    let vv3 = (spv3 as i128).wrapping_shl(21).wrapping_add(sv3);

    // Range reduction - Step 4
    // Output range: vv4 in [-0x1.0002143p-29 , 0x1p-29]
    // idx4 = trunc(2^21 * (v + 65*2^-28 + 2^-29))
    let idx4 = ((((vv3 >> 68) as i32).wrapping_add(131)) >> 1) as usize;

    let z4 = log_table[3][idx4];
    sum = sum + z4;

    let s4: i128 = S4[idx4] as i128; // |s| < 2^-21, ulp = 2^-28
    // |s + v| < 2^-28, ulp = 2^-97
    let spv4 = s4.wrapping_shl(69).wrapping_add(vv3);
    // |s*v| < 2^-42, ulp = 2^(-97-28) = 2^-125
    let sv4 = s4 * vv3;
    // |vv4| < 2^-28, ulp = 2^-125
    let vv4 = spv4.wrapping_shl(28).wrapping_add(sv4);

    let mut z0 = if vv4 < 0 {
        DyadicFloat128 {
            sign: DyadicSign::Neg,
            exponent: -125,
            mantissa: (-vv4) as u128,
        }
    } else {
        DyadicFloat128 {
            sign: DyadicSign::Pos,
            exponent: -125,
            mantissa: vv4 as u128,
        }
    };
    z0.normalize();

    (z0, sum)
}