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Base: correctly rounded floats constructed from rationals
Constructing a floating-point number from a `Rational` should now be correctly rounded. Implementation approach: 1. Convert the (numerator, denominator) pair to a (sign bit, integral significand, exponent) triplet using integer arithmetic. The integer type in question must be wide enough. 2. Convert the above triplet into an instance of the chosen FP type. There is special support for IEEE 754 floating-point and for `BigFloat`, otherwise a fallback using `ldexp` is used. As a bonus, constructing a `BigFloat` from a `Rational` should now be thread-safe when the rounding mode and precision are provided to the constructor, because there is no access to the global precision or rounding mode settings. Updates #45213 Updates #50940 Updates #52507 Fixes #52394 Closes #52395 Fixes #52859
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| Original file line number | Diff line number | Diff line change |
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| # This file is a part of Julia. License is MIT: https://julialang.org/license | ||
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| module RationalToFloat | ||
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| const Rnd = Base.Rounding | ||
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| # Performance optimization. Unlike raw `<<` or `>>>`, this is supposed | ||
| # to compile to a single instruction, because the semantics correspond | ||
| # to what hardware usually provides. | ||
| function machine_shift(shift::S, a::T, b) where {S,T<:Base.BitInteger} | ||
| @inline begin | ||
| mask = 8*sizeof(T) - 1 | ||
| c = b & mask | ||
| shift(a, c) | ||
| end | ||
| end | ||
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| machine_shift(::S, a::Bool, ::Any) where {S} = error("unsupported") | ||
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| # Fallback for `BigInt` etc. | ||
| machine_shift(shift::S, a, b) where {S} = shift(a, b) | ||
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| # Arguments are positive integers. | ||
| function div_significand_with_remainder(num, den, minimum_significand_size) | ||
| clamped = x -> max(zero(x), x)::typeof(x) | ||
| bw = Base.top_set_bit # bit width | ||
| shift = clamped(minimum_significand_size + bw(den) - bw(num) + 0x2) | ||
| t = machine_shift(<<, num, shift) | ||
| (divrem(t, den, RoundToZero)..., shift) | ||
| end | ||
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| # `divrem(n, 1<<k, RoundToZero)` | ||
| function divrem_2(n, k) | ||
| quo = machine_shift(>>>, n, k) | ||
| tmp = machine_shift(<<, quo, k) | ||
| rem = n - tmp | ||
| (quo, rem) | ||
| end | ||
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| function to_float_components_impl(num, den, precision, max_subnormal_exp) | ||
| # `+1` because we need an extra, "round", bit for some rounding modes. | ||
| # | ||
| # TODO: as a performance optimization, only do this when required | ||
| # by the rounding mode | ||
| prec_p_1 = precision + true | ||
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| (quo0, rem0, shift) = div_significand_with_remainder(num, den, prec_p_1) | ||
| width = Base.top_set_bit(quo0) | ||
| excess_width = width - prec_p_1 | ||
| exp = width - shift - true | ||
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| exp_underflow = if isnothing(max_subnormal_exp) | ||
| zero(exp) | ||
| else | ||
| let d = max_subnormal_exp - exp, T = typeof(d), z = zero(d)::T | ||
| (signbit(d) ? z : d + true)::T | ||
| end | ||
| end | ||
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| (quo1, rem1) = divrem_2(quo0, exp_underflow + excess_width) | ||
| integral_significand = quo1 >>> true | ||
| round_bit = quo1 % Bool | ||
| sticky_bit = !iszero(rem1) | !iszero(rem0) | ||
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| (; integral_significand, exponent = exp, round_bit, sticky_bit) | ||
| end | ||
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| struct RoundingIncrementHelper | ||
| final_bit::Bool | ||
| round_bit::Bool | ||
| sticky_bit::Bool | ||
| end | ||
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| (h::RoundingIncrementHelper)(::Rnd.FinalBit) = h.final_bit | ||
| (h::RoundingIncrementHelper)(::Rnd.RoundBit) = h.round_bit | ||
| (h::RoundingIncrementHelper)(::Rnd.StickyBit) = h.sticky_bit | ||
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| function to_float_components_rounded(num, den, precision, max_subnormal_exp, romo, sign_bit) | ||
| overflows = (x, p) -> x == machine_shift(<<, one(x), p) | ||
| t = to_float_components_impl(num, den, precision, max_subnormal_exp) | ||
| raw_significand = t.integral_significand | ||
| rh = RoundingIncrementHelper(raw_significand % Bool, t.round_bit, t.sticky_bit) | ||
| incr = Rnd.correct_rounding_requires_increment(rh, romo, sign_bit) | ||
| rounded = raw_significand + incr | ||
| (integral_significand, exponent) = let exp = t.exponent | ||
| if overflows(rounded, precision) | ||
| (rounded >>> true, exp + true) | ||
| else | ||
| (rounded, exp) | ||
| end | ||
| end | ||
| (; integral_significand, exponent) | ||
| end | ||
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| function to_float_components(::Type{T}, num, den, precision, max_subnormal_exp, romo, sb) where {T} | ||
| to_float_components_rounded(abs(T(num)), den, precision, max_subnormal_exp, romo, sb) | ||
| end | ||
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| function to_floating_point_fallback(::Type{T}, ::Type{S}, num, den, rm, prec) where {T,S} | ||
| num_is_zero = iszero(num) | ||
| den_is_zero = iszero(den) | ||
| sb = signbit(num) | ||
| is_zero = num_is_zero & !den_is_zero | ||
| is_inf = !num_is_zero & den_is_zero | ||
| is_regular = !num_is_zero & !den_is_zero | ||
| if is_regular | ||
| let | ||
| c = to_float_components(S, num, den, prec, nothing, rm, sb) | ||
| exp = c.exponent | ||
| signif = T(c.integral_significand)::T | ||
| let x = ldexp(signif, exp - prec + true)::T | ||
| sb ? -x : x | ||
| end::T | ||
| end | ||
| else | ||
| if is_zero | ||
| zero(T)::T | ||
| elseif is_inf | ||
| T(Inf)::T | ||
| else | ||
| T(NaN)::T | ||
| end | ||
| end::T | ||
| end | ||
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| function to_floating_point_impl(::Type{T}, ::Type{S}, num, den, rm, prec) where {T,S} | ||
| to_floating_point_fallback(T, S, num, den, rm, prec) | ||
| end | ||
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| function to_floating_point_impl(::Type{T}, ::Type{S}, num, den, rm, prec) where {T<:Base.IEEEFloat,S} | ||
| num_is_zero = iszero(num) | ||
| den_is_zero = iszero(den) | ||
| sb = signbit(num) | ||
| is_zero = num_is_zero & !den_is_zero | ||
| is_inf = !num_is_zero & den_is_zero | ||
| is_regular = !num_is_zero & !den_is_zero | ||
| (rm_is_to_zero, rm_is_from_zero) = if Rnd.rounds_to_nearest(rm) | ||
| (false, false) | ||
| else | ||
| let from = Rnd.rounds_away_from_zero(rm, sb) | ||
| (!from, from) | ||
| end | ||
| end::NTuple{2,Bool} | ||
| exp_max = Base.exponent_max(T) | ||
| exp_min = Base.exponent_min(T) | ||
| ieee_repr = Base.ieee754_representation | ||
| repr_zero = ieee_repr(T, sb, Val(:zero)) | ||
| repr_inf = ieee_repr(T, sb, Val(:inf)) | ||
| repr_nan = ieee_repr(T, sb, Val(:nan)) | ||
| U = typeof(repr_zero) | ||
| repr_zero::U | ||
| repr_inf::U | ||
| repr_nan::U | ||
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| ret_u = if is_regular | ||
| let | ||
| c = let e = exp_min - 1 | ||
| to_float_components(S, num, den, prec, e, rm, sb) | ||
| end | ||
| exp = c.exponent | ||
| exp_diff = exp - exp_min | ||
| is_normal = 0 ≤ exp_diff | ||
| exp_is_huge_p = exp_max < exp | ||
| exp_is_huge_n = signbit(exp_diff + prec) | ||
| rounds_to_inf = exp_is_huge_p & !rm_is_to_zero | ||
| rounds_to_zero = exp_is_huge_n & !rm_is_from_zero | ||
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| if !rounds_to_zero & !exp_is_huge_p | ||
| let signif = (c.integral_significand % U) & Base.significand_mask(T) | ||
| exp_field = (max(exp_diff, zero(exp_diff)) + is_normal) % U | ||
| ieee_repr(T, sb, exp_field, signif)::U | ||
| end | ||
| elseif rounds_to_zero | ||
| repr_zero | ||
| elseif rounds_to_inf | ||
| repr_inf | ||
| else | ||
| ieee_repr(T, sb, Val(:omega)) | ||
| end | ||
| end | ||
| else | ||
| if is_zero | ||
| repr_zero | ||
| elseif is_inf | ||
| repr_inf | ||
| else | ||
| repr_nan | ||
| end | ||
| end::U | ||
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| reinterpret(T, ret_u)::T | ||
| end | ||
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| # `BigInt` is a safe default. | ||
| to_float_promote_type(::Type{F}, ::Type{S}) where {F,S} = BigInt | ||
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| const BitIntegerOrBool = Union{Bool,Base.BitInteger} | ||
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| # As an optimization, use an integer type narrower than `BigInt` when possible. | ||
| function to_float_promote_type(::Type{F}, ::Type{S}) where {F<:Base.IEEEFloat,S<:BitIntegerOrBool} | ||
| Max = if sizeof(F) ≤ sizeof(S) | ||
| S | ||
| else | ||
| (S <: Signed) ? Base.inttype(F) : Base.uinttype(F) | ||
| end | ||
| widen(Max) | ||
| end | ||
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| function to_floating_point(::Type{F}, num::T, den::T, rm, prec) where {F,T} | ||
| S = to_float_promote_type(F, T) | ||
| to_floating_point_impl(F, S, num, den, rm, prec) | ||
| end | ||
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| end |
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