1//! Implementation of the Eisel-Lemire algorithm.
2//!
3//! This is adapted from [fast-float-rust](https://github.com/aldanor/fast-float-rust),
4//! a port of [fast_float](https://github.com/fastfloat/fast_float) to Rust.
5
6#![cfg(not(feature = "compact"))]
7#![doc(hidden)]
8
9use crate::extended_float::ExtendedFloat;
10use crate::num::Float;
11use crate::number::Number;
12use crate::table::{LARGEST_POWER_OF_FIVE, POWER_OF_FIVE_128, SMALLEST_POWER_OF_FIVE};
13
14/// Ensure truncation of digits doesn't affect our computation, by doing 2 passes.
15#[inline]
16pub fn lemire<F: Float>(num: &Number) -> ExtendedFloat {
17    // If significant digits were truncated, then we can have rounding error
18    // only if `mantissa + 1` produces a different result. We also avoid
19    // redundantly using the Eisel-Lemire algorithm if it was unable to
20    // correctly round on the first pass.
21    let mut fp = compute_float::<F>(num.exponent, num.mantissa);
22    if num.many_digits && fp.exp >= 0 && fp != compute_float::<F>(num.exponent, num.mantissa + 1) {
23        // Need to re-calculate, since the previous values are rounded
24        // when the slow path algorithm expects a normalized extended float.
25        fp = compute_error::<F>(num.exponent, num.mantissa);
26    }
27    fp
28}
29
30/// Compute a float using an extended-precision representation.
31///
32/// Fast conversion of a the significant digits and decimal exponent
33/// a float to a extended representation with a binary float. This
34/// algorithm will accurately parse the vast majority of cases,
35/// and uses a 128-bit representation (with a fallback 192-bit
36/// representation).
37///
38/// This algorithm scales the exponent by the decimal exponent
39/// using pre-computed powers-of-5, and calculates if the
40/// representation can be unambiguously rounded to the nearest
41/// machine float. Near-halfway cases are not handled here,
42/// and are represented by a negative, biased binary exponent.
43///
44/// The algorithm is described in detail in "Daniel Lemire, Number Parsing
45/// at a Gigabyte per Second" in section 5, "Fast Algorithm", and
46/// section 6, "Exact Numbers And Ties", available online:
47/// <https://arxiv.org/abs/2101.11408.pdf>.
48pub fn compute_float<F: Float>(q: i32, mut w: u64) -> ExtendedFloat {
49    let fp_zero = ExtendedFloat {
50        mant: 0,
51        exp: 0,
52    };
53    let fp_inf = ExtendedFloat {
54        mant: 0,
55        exp: F::INFINITE_POWER,
56    };
57
58    // Short-circuit if the value can only be a literal 0 or infinity.
59    if w == 0 || q < F::SMALLEST_POWER_OF_TEN {
60        return fp_zero;
61    } else if q > F::LARGEST_POWER_OF_TEN {
62        return fp_inf;
63    }
64    // Normalize our significant digits, so the most-significant bit is set.
65    let lz = w.leading_zeros() as i32;
66    w <<= lz;
67    let (lo, hi) = compute_product_approx(q, w, F::MANTISSA_SIZE as usize + 3);
68    if lo == 0xFFFF_FFFF_FFFF_FFFF {
69        // If we have failed to approximate w x 5^-q with our 128-bit value.
70        // Since the addition of 1 could lead to an overflow which could then
71        // round up over the half-way point, this can lead to improper rounding
72        // of a float.
73        //
74        // However, this can only occur if q ∈ [-27, 55]. The upper bound of q
75        // is 55 because 5^55 < 2^128, however, this can only happen if 5^q > 2^64,
76        // since otherwise the product can be represented in 64-bits, producing
77        // an exact result. For negative exponents, rounding-to-even can
78        // only occur if 5^-q < 2^64.
79        //
80        // For detailed explanations of rounding for negative exponents, see
81        // <https://arxiv.org/pdf/2101.11408.pdf#section.9.1>. For detailed
82        // explanations of rounding for positive exponents, see
83        // <https://arxiv.org/pdf/2101.11408.pdf#section.8>.
84        let inside_safe_exponent = (q >= -27) && (q <= 55);
85        if !inside_safe_exponent {
86            return compute_error_scaled::<F>(q, hi, lz);
87        }
88    }
89    let upperbit = (hi >> 63) as i32;
90    let mut mantissa = hi >> (upperbit + 64 - F::MANTISSA_SIZE - 3);
91    let mut power2 = power(q) + upperbit - lz - F::MINIMUM_EXPONENT;
92    if power2 <= 0 {
93        if -power2 + 1 >= 64 {
94            // Have more than 64 bits below the minimum exponent, must be 0.
95            return fp_zero;
96        }
97        // Have a subnormal value.
98        mantissa >>= -power2 + 1;
99        mantissa += mantissa & 1;
100        mantissa >>= 1;
101        power2 = (mantissa >= (1_u64 << F::MANTISSA_SIZE)) as i32;
102        return ExtendedFloat {
103            mant: mantissa,
104            exp: power2,
105        };
106    }
107    // Need to handle rounding ties. Normally, we need to round up,
108    // but if we fall right in between and and we have an even basis, we
109    // need to round down.
110    //
111    // This will only occur if:
112    //  1. The lower 64 bits of the 128-bit representation is 0.
113    //      IE, 5^q fits in single 64-bit word.
114    //  2. The least-significant bit prior to truncated mantissa is odd.
115    //  3. All the bits truncated when shifting to mantissa bits + 1 are 0.
116    //
117    // Or, we may fall between two floats: we are exactly halfway.
118    if lo <= 1
119        && q >= F::MIN_EXPONENT_ROUND_TO_EVEN
120        && q <= F::MAX_EXPONENT_ROUND_TO_EVEN
121        && mantissa & 3 == 1
122        && (mantissa << (upperbit + 64 - F::MANTISSA_SIZE - 3)) == hi
123    {
124        // Zero the lowest bit, so we don't round up.
125        mantissa &= !1_u64;
126    }
127    // Round-to-even, then shift the significant digits into place.
128    mantissa += mantissa & 1;
129    mantissa >>= 1;
130    if mantissa >= (2_u64 << F::MANTISSA_SIZE) {
131        // Rounding up overflowed, so the carry bit is set. Set the
132        // mantissa to 1 (only the implicit, hidden bit is set) and
133        // increase the exponent.
134        mantissa = 1_u64 << F::MANTISSA_SIZE;
135        power2 += 1;
136    }
137    // Zero out the hidden bit.
138    mantissa &= !(1_u64 << F::MANTISSA_SIZE);
139    if power2 >= F::INFINITE_POWER {
140        // Exponent is above largest normal value, must be infinite.
141        return fp_inf;
142    }
143    ExtendedFloat {
144        mant: mantissa,
145        exp: power2,
146    }
147}
148
149/// Fallback algorithm to calculate the non-rounded representation.
150/// This calculates the extended representation, and then normalizes
151/// the resulting representation, so the high bit is set.
152#[inline]
153pub fn compute_error<F: Float>(q: i32, mut w: u64) -> ExtendedFloat {
154    let lz = w.leading_zeros() as i32;
155    w <<= lz;
156    let hi = compute_product_approx(q, w, F::MANTISSA_SIZE as usize + 3).1;
157    compute_error_scaled::<F>(q, hi, lz)
158}
159
160/// Compute the error from a mantissa scaled to the exponent.
161#[inline]
162pub fn compute_error_scaled<F: Float>(q: i32, mut w: u64, lz: i32) -> ExtendedFloat {
163    // Want to normalize the float, but this is faster than ctlz on most architectures.
164    let hilz = (w >> 63) as i32 ^ 1;
165    w <<= hilz;
166    let power2 = power(q as i32) + F::EXPONENT_BIAS - hilz - lz - 62;
167
168    ExtendedFloat {
169        mant: w,
170        exp: power2 + F::INVALID_FP,
171    }
172}
173
174/// Calculate a base 2 exponent from a decimal exponent.
175/// This uses a pre-computed integer approximation for
176/// log2(10), where 217706 / 2^16 is accurate for the
177/// entire range of non-finite decimal exponents.
178#[inline]
179fn power(q: i32) -> i32 {
180    (q.wrapping_mul(152_170 + 65536) >> 16) + 63
181}
182
183#[inline]
184fn full_multiplication(a: u64, b: u64) -> (u64, u64) {
185    let r = (a as u128) * (b as u128);
186    (r as u64, (r >> 64) as u64)
187}
188
189// This will compute or rather approximate w * 5**q and return a pair of 64-bit words
190// approximating the result, with the "high" part corresponding to the most significant
191// bits and the low part corresponding to the least significant bits.
192fn compute_product_approx(q: i32, w: u64, precision: usize) -> (u64, u64) {
193    debug_assert!(q >= SMALLEST_POWER_OF_FIVE);
194    debug_assert!(q <= LARGEST_POWER_OF_FIVE);
195    debug_assert!(precision <= 64);
196
197    let mask = if precision < 64 {
198        0xFFFF_FFFF_FFFF_FFFF_u64 >> precision
199    } else {
200        0xFFFF_FFFF_FFFF_FFFF_u64
201    };
202
203    // 5^q < 2^64, then the multiplication always provides an exact value.
204    // That means whenever we need to round ties to even, we always have
205    // an exact value.
206    let index = (q - SMALLEST_POWER_OF_FIVE) as usize;
207    let (lo5, hi5) = POWER_OF_FIVE_128[index];
208    // Only need one multiplication as long as there is 1 zero but
209    // in the explicit mantissa bits, +1 for the hidden bit, +1 to
210    // determine the rounding direction, +1 for if the computed
211    // product has a leading zero.
212    let (mut first_lo, mut first_hi) = full_multiplication(w, lo5);
213    if first_hi & mask == mask {
214        // Need to do a second multiplication to get better precision
215        // for the lower product. This will always be exact
216        // where q is < 55, since 5^55 < 2^128. If this wraps,
217        // then we need to need to round up the hi product.
218        let (_, second_hi) = full_multiplication(w, hi5);
219        first_lo = first_lo.wrapping_add(second_hi);
220        if second_hi > first_lo {
221            first_hi += 1;
222        }
223    }
224    (first_lo, first_hi)
225}
226