core/num/dec2flt/
lemire.rs

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
164
165
166
167
168
//! Implementation of the Eisel-Lemire algorithm.

use crate::num::dec2flt::common::BiasedFp;
use crate::num::dec2flt::float::RawFloat;
use crate::num::dec2flt::table::{
    LARGEST_POWER_OF_FIVE, POWER_OF_FIVE_128, SMALLEST_POWER_OF_FIVE,
};

/// Compute w * 10^q using an extended-precision float representation.
///
/// Fast conversion of a the significant digits and decimal exponent
/// a float to an extended representation with a binary float. This
/// algorithm will accurately parse the vast majority of cases,
/// and uses a 128-bit representation (with a fallback 192-bit
/// representation).
///
/// This algorithm scales the exponent by the decimal exponent
/// using pre-computed powers-of-5, and calculates if the
/// representation can be unambiguously rounded to the nearest
/// machine float. Near-halfway cases are not handled here,
/// and are represented by a negative, biased binary exponent.
///
/// The algorithm is described in detail in "Daniel Lemire, Number Parsing
/// at a Gigabyte per Second" in section 5, "Fast Algorithm", and
/// section 6, "Exact Numbers And Ties", available online:
/// <https://arxiv.org/abs/2101.11408.pdf>.
pub fn compute_float<F: RawFloat>(q: i64, mut w: u64) -> BiasedFp {
    let fp_zero = BiasedFp::zero_pow2(0);
    let fp_inf = BiasedFp::zero_pow2(F::INFINITE_POWER);
    let fp_error = BiasedFp::zero_pow2(-1);

    // Short-circuit if the value can only be a literal 0 or infinity.
    if w == 0 || q < F::SMALLEST_POWER_OF_TEN as i64 {
        return fp_zero;
    } else if q > F::LARGEST_POWER_OF_TEN as i64 {
        return fp_inf;
    }
    // Normalize our significant digits, so the most-significant bit is set.
    let lz = w.leading_zeros();
    w <<= lz;
    let (lo, hi) = compute_product_approx(q, w, F::MANTISSA_EXPLICIT_BITS + 3);
    if lo == 0xFFFF_FFFF_FFFF_FFFF {
        // If we have failed to approximate w x 5^-q with our 128-bit value.
        // Since the addition of 1 could lead to an overflow which could then
        // round up over the half-way point, this can lead to improper rounding
        // of a float.
        //
        // However, this can only occur if q ∈ [-27, 55]. The upper bound of q
        // is 55 because 5^55 < 2^128, however, this can only happen if 5^q > 2^64,
        // since otherwise the product can be represented in 64-bits, producing
        // an exact result. For negative exponents, rounding-to-even can
        // only occur if 5^-q < 2^64.
        //
        // For detailed explanations of rounding for negative exponents, see
        // <https://arxiv.org/pdf/2101.11408.pdf#section.9.1>. For detailed
        // explanations of rounding for positive exponents, see
        // <https://arxiv.org/pdf/2101.11408.pdf#section.8>.
        let inside_safe_exponent = (q >= -27) && (q <= 55);
        if !inside_safe_exponent {
            return fp_error;
        }
    }
    let upperbit = (hi >> 63) as i32;
    let mut mantissa = hi >> (upperbit + 64 - F::MANTISSA_EXPLICIT_BITS as i32 - 3);
    let mut power2 = power(q as i32) + upperbit - lz as i32 - F::MINIMUM_EXPONENT;
    if power2 <= 0 {
        if -power2 + 1 >= 64 {
            // Have more than 64 bits below the minimum exponent, must be 0.
            return fp_zero;
        }
        // Have a subnormal value.
        mantissa >>= -power2 + 1;
        mantissa += mantissa & 1;
        mantissa >>= 1;
        power2 = (mantissa >= (1_u64 << F::MANTISSA_EXPLICIT_BITS)) as i32;
        return BiasedFp { f: mantissa, e: power2 };
    }
    // Need to handle rounding ties. Normally, we need to round up,
    // but if we fall right in between and we have an even basis, we
    // need to round down.
    //
    // This will only occur if:
    //  1. The lower 64 bits of the 128-bit representation is 0.
    //      IE, 5^q fits in single 64-bit word.
    //  2. The least-significant bit prior to truncated mantissa is odd.
    //  3. All the bits truncated when shifting to mantissa bits + 1 are 0.
    //
    // Or, we may fall between two floats: we are exactly halfway.
    if lo <= 1
        && q >= F::MIN_EXPONENT_ROUND_TO_EVEN as i64
        && q <= F::MAX_EXPONENT_ROUND_TO_EVEN as i64
        && mantissa & 3 == 1
        && (mantissa << (upperbit + 64 - F::MANTISSA_EXPLICIT_BITS as i32 - 3)) == hi
    {
        // Zero the lowest bit, so we don't round up.
        mantissa &= !1_u64;
    }
    // Round-to-even, then shift the significant digits into place.
    mantissa += mantissa & 1;
    mantissa >>= 1;
    if mantissa >= (2_u64 << F::MANTISSA_EXPLICIT_BITS) {
        // Rounding up overflowed, so the carry bit is set. Set the
        // mantissa to 1 (only the implicit, hidden bit is set) and
        // increase the exponent.
        mantissa = 1_u64 << F::MANTISSA_EXPLICIT_BITS;
        power2 += 1;
    }
    // Zero out the hidden bit.
    mantissa &= !(1_u64 << F::MANTISSA_EXPLICIT_BITS);
    if power2 >= F::INFINITE_POWER {
        // Exponent is above largest normal value, must be infinite.
        return fp_inf;
    }
    BiasedFp { f: mantissa, e: power2 }
}

/// Calculate a base 2 exponent from a decimal exponent.
/// This uses a pre-computed integer approximation for
/// log2(10), where 217706 / 2^16 is accurate for the
/// entire range of non-finite decimal exponents.
#[inline]
fn power(q: i32) -> i32 {
    (q.wrapping_mul(152_170 + 65536) >> 16) + 63
}

#[inline]
fn full_multiplication(a: u64, b: u64) -> (u64, u64) {
    let r = (a as u128) * (b as u128);
    (r as u64, (r >> 64) as u64)
}

// This will compute or rather approximate w * 5**q and return a pair of 64-bit words
// approximating the result, with the "high" part corresponding to the most significant
// bits and the low part corresponding to the least significant bits.
fn compute_product_approx(q: i64, w: u64, precision: usize) -> (u64, u64) {
    debug_assert!(q >= SMALLEST_POWER_OF_FIVE as i64);
    debug_assert!(q <= LARGEST_POWER_OF_FIVE as i64);
    debug_assert!(precision <= 64);

    let mask = if precision < 64 {
        0xFFFF_FFFF_FFFF_FFFF_u64 >> precision
    } else {
        0xFFFF_FFFF_FFFF_FFFF_u64
    };

    // 5^q < 2^64, then the multiplication always provides an exact value.
    // That means whenever we need to round ties to even, we always have
    // an exact value.
    let index = (q - SMALLEST_POWER_OF_FIVE as i64) as usize;
    let (lo5, hi5) = POWER_OF_FIVE_128[index];
    // Only need one multiplication as long as there is 1 zero but
    // in the explicit mantissa bits, +1 for the hidden bit, +1 to
    // determine the rounding direction, +1 for if the computed
    // product has a leading zero.
    let (mut first_lo, mut first_hi) = full_multiplication(w, lo5);
    if first_hi & mask == mask {
        // Need to do a second multiplication to get better precision
        // for the lower product. This will always be exact
        // where q is < 55, since 5^55 < 2^128. If this wraps,
        // then we need to round up the hi product.
        let (_, second_hi) = full_multiplication(w, hi5);
        first_lo = first_lo.wrapping_add(second_hi);
        if second_hi > first_lo {
            first_hi += 1;
        }
    }
    (first_lo, first_hi)
}