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https://github.com/nlohmann/json
synced 2024-12-11 20:20:06 +00:00
332f352033
This is an attempt to fix #360. The algorithm produces decimal representations which are guaranteed to roundtrip and in ~99.8% actually produces the shortest possible representation. So this is a nice compromise between using a precision of digits10 and max_digits10. Note 1: The implementation only works for IEEE single/double precision numbers. So the old implementation is kept for compatibility with non-IEEE implementations and 'long double'. Note 2: If number_float_t is 'float', not all serialized numbers can be recovered using strtod (strtof works, though). (There is exactly one such number and the result is off by 1 ulp.) This can be avoided by changing the implementation (the fix is trivial), but then the resulting decimal numbers are not exactly short.
1028 lines
35 KiB
C++
1028 lines
35 KiB
C++
#pragma once
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#include <cassert> // assert
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#include <ciso646> // or, and, not
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#include <cmath> // signbit, isfinite
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#include <cstdint> // intN_t, uintN_t
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#include <cstring> // memcpy, memmove
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namespace nlohmann
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{
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namespace detail
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{
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// Implements the Grisu2 algorithm for binary to decimal floating-point conversion.
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//
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// This implementation is a slightly modified version of the reference implementation which may be
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// obtained from http://florian.loitsch.com/publications (bench.tar.gz).
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//
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// The code is distributed under the MIT license, Copyright (c) 2009 Florian Loitsch.
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//
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// For a detailed description of the algorithm see:
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//
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// [1] Loitsch, "Printing Floating-Point Numbers Quickly and Accurately with Integers",
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// Proceedings of the ACM SIGPLAN 2010 Conference on Programming Language Design and Implementation, PLDI 2010
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// [2] Burger, Dybvig, "Printing Floating-Point Numbers Quickly and Accurately",
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// Proceedings of the ACM SIGPLAN 1996 Conference on Programming Language Design and Implementation, PLDI 1996
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namespace dtoa_impl
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{
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template <typename Target, typename Source>
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Target reinterpret_bits(Source source)
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{
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static_assert(sizeof(Target) == sizeof(Source), "size mismatch");
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Target target;
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std::memcpy(&target, &source, sizeof(Source));
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return target;
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}
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struct diyfp // f * 2^e
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{
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static constexpr int kPrecision = 64; // = q
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uint64_t f;
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int e;
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constexpr diyfp() : f(0), e(0) {}
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constexpr diyfp(uint64_t f_, int e_) : f(f_), e(e_) {}
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// Returns x - y.
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// PRE: x.e == y.e and x.f >= y.f
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static diyfp sub(diyfp x, diyfp y);
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// Returns x * y.
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// The result is rounded. (Only the upper q bits are returned.)
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static diyfp mul(diyfp x, diyfp y);
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// Normalize x such that the significand is >= 2^(q-1).
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// PRE: x.f != 0
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static diyfp normalize(diyfp x);
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// Normalize x such that the result has the exponent E.
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// PRE: e >= x.e and the upper e - x.e bits of x.f must be zero.
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static diyfp normalize_to(diyfp x, int e);
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};
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inline diyfp diyfp::sub(diyfp x, diyfp y)
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{
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assert(x.e == y.e);
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assert(x.f >= y.f);
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return diyfp(x.f - y.f, x.e);
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}
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inline diyfp diyfp::mul(diyfp x, diyfp y)
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{
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static_assert(kPrecision == 64, "internal error");
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// Computes:
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// f = round((x.f * y.f) / 2^q)
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// e = x.e + y.e + q
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// Emulate the 64-bit * 64-bit multiplication:
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//
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// p = u * v
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// = (u_lo + 2^32 u_hi) (v_lo + 2^32 v_hi)
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// = (u_lo v_lo ) + 2^32 ((u_lo v_hi ) + (u_hi v_lo )) + 2^64 (u_hi v_hi )
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// = (p0 ) + 2^32 ((p1 ) + (p2 )) + 2^64 (p3 )
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// = (p0_lo + 2^32 p0_hi) + 2^32 ((p1_lo + 2^32 p1_hi) + (p2_lo + 2^32 p2_hi)) + 2^64 (p3 )
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// = (p0_lo ) + 2^32 (p0_hi + p1_lo + p2_lo ) + 2^64 (p1_hi + p2_hi + p3)
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// = (p0_lo ) + 2^32 (Q ) + 2^64 (H )
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// = (p0_lo ) + 2^32 (Q_lo + 2^32 Q_hi ) + 2^64 (H )
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//
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// (Since Q might be larger than 2^32 - 1)
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//
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// = (p0_lo + 2^32 Q_lo) + 2^64 (Q_hi + H)
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//
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// (Q_hi + H does not overflow a 64-bit int)
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//
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// = p_lo + 2^64 p_hi
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const uint64_t u_lo = x.f & 0xFFFFFFFF;
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const uint64_t u_hi = x.f >> 32;
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const uint64_t v_lo = y.f & 0xFFFFFFFF;
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const uint64_t v_hi = y.f >> 32;
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const uint64_t p0 = u_lo * v_lo;
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const uint64_t p1 = u_lo * v_hi;
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const uint64_t p2 = u_hi * v_lo;
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const uint64_t p3 = u_hi * v_hi;
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const uint64_t p0_hi = p0 >> 32;
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const uint64_t p1_lo = p1 & 0xFFFFFFFF;
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const uint64_t p1_hi = p1 >> 32;
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const uint64_t p2_lo = p2 & 0xFFFFFFFF;
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const uint64_t p2_hi = p2 >> 32;
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uint64_t Q = p0_hi + p1_lo + p2_lo;
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// The full product might now be computed as
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//
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// p_hi = p3 + p2_hi + p1_hi + (Q >> 32)
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// p_lo = p0_lo + (Q << 32)
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//
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// But in this particular case here, the full p_lo is not required.
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// Effectively we only need to add the highest bit in p_lo to p_hi (and
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// Q_hi + 1 does not overflow).
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Q += uint64_t{1} << (64 - 32 - 1); // round, ties up
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const uint64_t h = p3 + p2_hi + p1_hi + (Q >> 32);
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return diyfp(h, x.e + y.e + 64);
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}
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inline diyfp diyfp::normalize(diyfp x)
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{
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assert(x.f != 0);
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while ((x.f >> 63) == 0)
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{
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x.f <<= 1;
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x.e--;
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}
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return x;
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}
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inline diyfp diyfp::normalize_to(diyfp x, int target_exponent)
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{
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const int delta = x.e - target_exponent;
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assert(delta >= 0);
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assert(((x.f << delta) >> delta) == x.f);
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return diyfp(x.f << delta, target_exponent);
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}
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struct boundaries
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{
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diyfp w;
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diyfp minus;
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diyfp plus;
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};
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// Compute the (normalized) diyfp representing the input number 'value' and its boundaries.
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// PRE: value must be finite and positive
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template <typename FloatType>
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boundaries compute_boundaries(FloatType value)
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{
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assert(std::isfinite(value));
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assert(value > 0);
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// Convert the IEEE representation into a diyfp.
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//
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// If v is denormal:
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// value = 0.F * 2^(1 - bias) = ( F) * 2^(1 - bias - (p-1))
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// If v is normalized:
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// value = 1.F * 2^(E - bias) = (2^(p-1) + F) * 2^(E - bias - (p-1))
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static_assert(std::numeric_limits<FloatType>::is_iec559,
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"internal error: dtoa_short requires an IEEE-754 floating-point implementation");
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constexpr int kPrecision = std::numeric_limits<FloatType>::digits; // = p (includes the hidden bit)
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constexpr int kBias = std::numeric_limits<FloatType>::max_exponent - 1 + (kPrecision - 1);
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constexpr int kMinExp = 1 - kBias;
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constexpr uint64_t kHiddenBit = uint64_t{1} << (kPrecision - 1); // = 2^(p-1)
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using bits_type = typename std::conditional< kPrecision == 24, uint32_t, uint64_t >::type;
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const uint64_t bits = reinterpret_bits<bits_type>(value);
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const uint64_t E = bits >> (kPrecision - 1);
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const uint64_t F = bits & (kHiddenBit - 1);
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const bool is_denormal = (E == 0);
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const diyfp v
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= is_denormal
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? diyfp(F, 1 - kBias)
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: diyfp(F + kHiddenBit, static_cast<int>(E) - kBias);
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// Compute the boundaries m- and m+ of the floating-point value
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// v = f * 2^e.
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//
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// Determine v- and v+, the floating-point predecessor and successor if v,
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// respectively.
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//
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// v- = v - 2^e if f != 2^(p-1) or e == e_min (A)
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// = v - 2^(e-1) if f == 2^(p-1) and e > e_min (B)
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//
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// v+ = v + 2^e
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//
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// Let m- = (v- + v) / 2 and m+ = (v + v+) / 2. All real numbers _strictly_
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// between m- and m+ round to v, regardless of how the input rounding
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// algorithm breaks ties.
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//
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// ---+-------------+-------------+-------------+-------------+--- (A)
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// v- m- v m+ v+
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//
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// -----------------+------+------+-------------+-------------+--- (B)
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// v- m- v m+ v+
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const bool lower_boundary_is_closer = (F == 0 and E > 1);
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const diyfp m_plus = diyfp(2 * v.f + 1, v.e - 1);
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const diyfp m_minus
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= lower_boundary_is_closer
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? diyfp(4 * v.f - 1, v.e - 2) // (B)
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: diyfp(2 * v.f - 1, v.e - 1); // (A)
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// Determine the normalized w+ = m+.
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const diyfp w_plus = diyfp::normalize(m_plus);
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// Determine w- = m- such that e_(w-) = e_(w+).
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const diyfp w_minus = diyfp::normalize_to(m_minus, w_plus.e);
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return {diyfp::normalize(v), w_minus, w_plus};
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}
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// Given normalized diyfp w, Grisu needs to find a (normalized) cached
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// power-of-ten c, such that the exponent of the product c * w = f * 2^e lies
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// within a certain range [alpha, gamma] (Definition 3.2 from [1])
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//
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// alpha <= e = e_c + e_w + q <= gamma
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//
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// or
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//
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// f_c * f_w * 2^alpha <= f_c 2^(e_c) * f_w 2^(e_w) * 2^q
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// <= f_c * f_w * 2^gamma
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//
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// Since c and w are normalized, i.e. 2^(q-1) <= f < 2^q, this implies
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//
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// 2^(q-1) * 2^(q-1) * 2^alpha <= c * w * 2^q < 2^q * 2^q * 2^gamma
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//
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// or
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//
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// 2^(q - 2 + alpha) <= c * w < 2^(q + gamma)
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//
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// The choice of (alpha,gamma) determines the size of the table and the form of
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// the digit generation procedure. Using (alpha,gamma)=(-60,-32) works out well
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// in practice:
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//
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// The idea is to cut the number c * w = f * 2^e into two parts, which can be
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// processed independently: An integral part p1, and a fractional part p2:
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//
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// f * 2^e = ( (f div 2^-e) * 2^-e + (f mod 2^-e) ) * 2^e
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// = (f div 2^-e) + (f mod 2^-e) * 2^e
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// = p1 + p2 * 2^e
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//
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// The conversion of p1 into decimal form requires a series of divisions and
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// modulos by (a power of) 10. These operations are faster for 32-bit than for
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// 64-bit integers, so p1 should ideally fit into a 32-bit integer. This can be
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// achieved by choosing
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//
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// -e >= 32 or e <= -32 := gamma
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//
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// In order to convert the fractional part
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//
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// p2 * 2^e = p2 / 2^-e = d[-1] / 10^1 + d[-2] / 10^2 + ...
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//
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// into decimal form, the fraction is repeatedly multiplied by 10 and the digits
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// d[-i] are extracted in order:
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//
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// (10 * p2) div 2^-e = d[-1]
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// (10 * p2) mod 2^-e = d[-2] / 10^1 + ...
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//
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// The multiplication by 10 must not overflow. It is sufficient to choose
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//
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// 10 * p2 < 16 * p2 = 2^4 * p2 <= 2^64.
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//
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// Since p2 = f mod 2^-e < 2^-e,
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//
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// -e <= 60 or e >= -60 := alpha
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constexpr int kAlpha = -60;
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constexpr int kGamma = -32;
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struct cached_power // c = f * 2^e ~= 10^k
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{
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uint64_t f;
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int e;
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int k;
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};
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// For a normalized diyfp w = f * 2^e, this function returns a (normalized)
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// cached power-of-ten c = f_c * 2^e_c, such that the exponent of the product
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// w * c satisfies (Definition 3.2 from [1])
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//
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// alpha <= e_c + e + q <= gamma.
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//
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inline cached_power get_cached_power_for_binary_exponent(int e)
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{
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// Now
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//
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// alpha <= e_c + e + q <= gamma (1)
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// ==> f_c * 2^alpha <= c * 2^e * 2^q
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//
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// and since the c's are normalized, 2^(q-1) <= f_c,
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//
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// ==> 2^(q - 1 + alpha) <= c * 2^(e + q)
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// ==> 2^(alpha - e - 1) <= c
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//
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// If c were an exakt power of ten, i.e. c = 10^k, one may determine k as
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//
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// k = ceil( log_10( 2^(alpha - e - 1) ) )
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// = ceil( (alpha - e - 1) * log_10(2) )
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//
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// From the paper:
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// "In theory the result of the procedure could be wrong since c is rounded,
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// and the computation itself is approximated [...]. In practice, however,
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// this simple function is sufficient."
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//
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// For IEEE double precision floating-point numbers converted into
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// normalized diyfp's w = f * 2^e, with q = 64,
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//
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// e >= -1022 (min IEEE exponent)
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// -52 (p - 1)
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// -52 (p - 1, possibly normalize denormal IEEE numbers)
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// -11 (normalize the diyfp)
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// = -1137
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//
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// and
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//
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// e <= +1023 (max IEEE exponent)
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// -52 (p - 1)
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// -11 (normalize the diyfp)
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// = 960
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//
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// This binary exponent range [-1137,960] results in a decimal exponent
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// range [-307,324]. One does not need to store a cached power for each
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// k in this range. For each such k it suffices to find a cached power
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// such that the exponent of the product lies in [alpha,gamma].
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// This implies that the difference of the decimal exponents of adjacent
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// table entries must be less than or equal to
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//
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// floor( (gamma - alpha) * log_10(2) ) = 8.
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//
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// (A smaller distance gamma-alpha would require a larger table.)
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// NB:
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// Actually this function returns c, such that -60 <= e_c + e + 64 <= -34.
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constexpr int kCachedPowersSize = 79;
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constexpr int kCachedPowersMinDecExp = -300;
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constexpr int kCachedPowersDecStep = 8;
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static constexpr cached_power kCachedPowers[] =
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{
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{ 0xAB70FE17C79AC6CA, -1060, -300 },
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{ 0xFF77B1FCBEBCDC4F, -1034, -292 },
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{ 0xBE5691EF416BD60C, -1007, -284 },
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{ 0x8DD01FAD907FFC3C, -980, -276 },
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{ 0xD3515C2831559A83, -954, -268 },
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{ 0x9D71AC8FADA6C9B5, -927, -260 },
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{ 0xEA9C227723EE8BCB, -901, -252 },
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{ 0xAECC49914078536D, -874, -244 },
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{ 0x823C12795DB6CE57, -847, -236 },
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{ 0xC21094364DFB5637, -821, -228 },
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{ 0x9096EA6F3848984F, -794, -220 },
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{ 0xD77485CB25823AC7, -768, -212 },
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{ 0xA086CFCD97BF97F4, -741, -204 },
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{ 0xEF340A98172AACE5, -715, -196 },
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{ 0xB23867FB2A35B28E, -688, -188 },
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{ 0x84C8D4DFD2C63F3B, -661, -180 },
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{ 0xC5DD44271AD3CDBA, -635, -172 },
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{ 0x936B9FCEBB25C996, -608, -164 },
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{ 0xDBAC6C247D62A584, -582, -156 },
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{ 0xA3AB66580D5FDAF6, -555, -148 },
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{ 0xF3E2F893DEC3F126, -529, -140 },
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{ 0xB5B5ADA8AAFF80B8, -502, -132 },
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{ 0x87625F056C7C4A8B, -475, -124 },
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{ 0xC9BCFF6034C13053, -449, -116 },
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{ 0x964E858C91BA2655, -422, -108 },
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{ 0xDFF9772470297EBD, -396, -100 },
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{ 0xA6DFBD9FB8E5B88F, -369, -92 },
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{ 0xF8A95FCF88747D94, -343, -84 },
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{ 0xB94470938FA89BCF, -316, -76 },
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{ 0x8A08F0F8BF0F156B, -289, -68 },
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{ 0xCDB02555653131B6, -263, -60 },
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{ 0x993FE2C6D07B7FAC, -236, -52 },
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{ 0xE45C10C42A2B3B06, -210, -44 },
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{ 0xAA242499697392D3, -183, -36 },
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{ 0xFD87B5F28300CA0E, -157, -28 },
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{ 0xBCE5086492111AEB, -130, -20 },
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{ 0x8CBCCC096F5088CC, -103, -12 },
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{ 0xD1B71758E219652C, -77, -4 },
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{ 0x9C40000000000000, -50, 4 },
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{ 0xE8D4A51000000000, -24, 12 },
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{ 0xAD78EBC5AC620000, 3, 20 },
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{ 0x813F3978F8940984, 30, 28 },
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{ 0xC097CE7BC90715B3, 56, 36 },
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{ 0x8F7E32CE7BEA5C70, 83, 44 },
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{ 0xD5D238A4ABE98068, 109, 52 },
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{ 0x9F4F2726179A2245, 136, 60 },
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{ 0xED63A231D4C4FB27, 162, 68 },
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{ 0xB0DE65388CC8ADA8, 189, 76 },
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{ 0x83C7088E1AAB65DB, 216, 84 },
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{ 0xC45D1DF942711D9A, 242, 92 },
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{ 0x924D692CA61BE758, 269, 100 },
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{ 0xDA01EE641A708DEA, 295, 108 },
|
|
{ 0xA26DA3999AEF774A, 322, 116 },
|
|
{ 0xF209787BB47D6B85, 348, 124 },
|
|
{ 0xB454E4A179DD1877, 375, 132 },
|
|
{ 0x865B86925B9BC5C2, 402, 140 },
|
|
{ 0xC83553C5C8965D3D, 428, 148 },
|
|
{ 0x952AB45CFA97A0B3, 455, 156 },
|
|
{ 0xDE469FBD99A05FE3, 481, 164 },
|
|
{ 0xA59BC234DB398C25, 508, 172 },
|
|
{ 0xF6C69A72A3989F5C, 534, 180 },
|
|
{ 0xB7DCBF5354E9BECE, 561, 188 },
|
|
{ 0x88FCF317F22241E2, 588, 196 },
|
|
{ 0xCC20CE9BD35C78A5, 614, 204 },
|
|
{ 0x98165AF37B2153DF, 641, 212 },
|
|
{ 0xE2A0B5DC971F303A, 667, 220 },
|
|
{ 0xA8D9D1535CE3B396, 694, 228 },
|
|
{ 0xFB9B7CD9A4A7443C, 720, 236 },
|
|
{ 0xBB764C4CA7A44410, 747, 244 },
|
|
{ 0x8BAB8EEFB6409C1A, 774, 252 },
|
|
{ 0xD01FEF10A657842C, 800, 260 },
|
|
{ 0x9B10A4E5E9913129, 827, 268 },
|
|
{ 0xE7109BFBA19C0C9D, 853, 276 },
|
|
{ 0xAC2820D9623BF429, 880, 284 },
|
|
{ 0x80444B5E7AA7CF85, 907, 292 },
|
|
{ 0xBF21E44003ACDD2D, 933, 300 },
|
|
{ 0x8E679C2F5E44FF8F, 960, 308 },
|
|
{ 0xD433179D9C8CB841, 986, 316 },
|
|
{ 0x9E19DB92B4E31BA9, 1013, 324 },
|
|
};
|
|
|
|
// This computation gives exactly the same results for k as
|
|
// k = ceil((kAlpha - e - 1) * 0.30102999566398114)
|
|
// for |e| <= 1500, but doesn't require floating-point operations.
|
|
// NB: log_10(2) ~= 78913 / 2^18
|
|
assert(e >= -1500);
|
|
assert(e <= 1500);
|
|
const int f = kAlpha - e - 1;
|
|
const int k = (f * 78913) / (1 << 18) + (f > 0);
|
|
|
|
const int index = (-kCachedPowersMinDecExp + k + (kCachedPowersDecStep - 1)) / kCachedPowersDecStep;
|
|
assert(index >= 0);
|
|
assert(index < kCachedPowersSize);
|
|
static_cast<void>(kCachedPowersSize); // Fix warning.
|
|
|
|
const cached_power cached = kCachedPowers[index];
|
|
assert(kAlpha <= cached.e + e + 64);
|
|
assert(kGamma >= cached.e + e + 64);
|
|
|
|
return cached;
|
|
}
|
|
|
|
// For n != 0, returns k, such that pow10 := 10^(k-1) <= n < 10^k.
|
|
// For n == 0, returns 1 and sets pow10 := 1.
|
|
inline int find_largest_pow10(uint32_t n, uint32_t& pow10)
|
|
{
|
|
if (n >= 1000000000) { pow10 = 1000000000; return 10; }
|
|
if (n >= 100000000) { pow10 = 100000000; return 9; }
|
|
if (n >= 10000000) { pow10 = 10000000; return 8; }
|
|
if (n >= 1000000) { pow10 = 1000000; return 7; }
|
|
if (n >= 100000) { pow10 = 100000; return 6; }
|
|
if (n >= 10000) { pow10 = 10000; return 5; }
|
|
if (n >= 1000) { pow10 = 1000; return 4; }
|
|
if (n >= 100) { pow10 = 100; return 3; }
|
|
if (n >= 10) { pow10 = 10; return 2; }
|
|
|
|
pow10 = 1; return 1;
|
|
}
|
|
|
|
inline void grisu2_round(char* buf, int len, uint64_t dist, uint64_t delta, uint64_t rest, uint64_t ten_k)
|
|
{
|
|
assert(len >= 1);
|
|
assert(dist <= delta);
|
|
assert(rest <= delta);
|
|
assert(ten_k > 0);
|
|
|
|
// <--------------------------- delta ---->
|
|
// <---- dist --------->
|
|
// --------------[------------------+-------------------]--------------
|
|
// M- w M+
|
|
//
|
|
// ten_k
|
|
// <------>
|
|
// <---- rest ---->
|
|
// --------------[------------------+----+--------------]--------------
|
|
// w V
|
|
// = buf * 10^k
|
|
//
|
|
// ten_k represents a unit-in-the-last-place in the decimal representation
|
|
// stored in buf.
|
|
// Decrement buf by ten_k while this takes buf closer to w.
|
|
|
|
// The tests are written in this order to avoid overflow in unsigned
|
|
// integer arithmetic.
|
|
|
|
while (rest < dist
|
|
and delta - rest >= ten_k
|
|
and (rest + ten_k < dist or dist - rest > rest + ten_k - dist))
|
|
{
|
|
assert(buf[len - 1] != '0');
|
|
buf[len - 1]--;
|
|
rest += ten_k;
|
|
}
|
|
}
|
|
|
|
// Generates V = buffer * 10^decimal_exponent, such that M- <= V <= M+.
|
|
// M- and M+ must be normalized and share the same exponent -60 <= e <= -32.
|
|
inline void grisu2_digit_gen(char* buffer, int& length, int& decimal_exponent, diyfp M_minus, diyfp w, diyfp M_plus)
|
|
{
|
|
static_assert(kAlpha >= -60, "internal error");
|
|
static_assert(kGamma <= -32, "internal error");
|
|
|
|
// Generates the digits (and the exponent) of a decimal floating-point
|
|
// number V = buffer * 10^decimal_exponent in the range [M-, M+]. The diyfp's
|
|
// w, M- and M+ share the same exponent e, which satisfies alpha <= e <= gamma.
|
|
//
|
|
// <--------------------------- delta ---->
|
|
// <---- dist --------->
|
|
// --------------[------------------+-------------------]--------------
|
|
// M- w M+
|
|
//
|
|
// Grisu2 generates the digits of M+ from left to right and stops as soon as
|
|
// V is in [M-,M+].
|
|
|
|
assert(M_plus.e >= kAlpha);
|
|
assert(M_plus.e <= kGamma);
|
|
|
|
uint64_t delta = diyfp::sub(M_plus, M_minus).f; // (significand of (M+ - M-), implicit exponent is e)
|
|
uint64_t dist = diyfp::sub(M_plus, w ).f; // (significand of (M+ - w ), implicit exponent is e)
|
|
|
|
// Split M+ = f * 2^e into two parts p1 and p2 (note: e < 0):
|
|
//
|
|
// M+ = f * 2^e
|
|
// = ((f div 2^-e) * 2^-e + (f mod 2^-e)) * 2^e
|
|
// = ((p1 ) * 2^-e + (p2 )) * 2^e
|
|
// = p1 + p2 * 2^e
|
|
|
|
const diyfp one(uint64_t{1} << -M_plus.e, M_plus.e);
|
|
|
|
uint32_t p1 = static_cast<uint32_t>(M_plus.f >> -one.e); // p1 = f div 2^-e (Since -e >= 32, p1 fits into a 32-bit int.)
|
|
uint64_t p2 = M_plus.f & (one.f - 1); // p2 = f mod 2^-e
|
|
|
|
// 1)
|
|
//
|
|
// Generate the digits of the integral part p1 = d[n-1]...d[1]d[0]
|
|
|
|
assert(p1 > 0);
|
|
|
|
uint32_t pow10;
|
|
const int k = find_largest_pow10(p1, pow10);
|
|
|
|
// 10^(k-1) <= p1 < 10^k, pow10 = 10^(k-1)
|
|
//
|
|
// p1 = (p1 div 10^(k-1)) * 10^(k-1) + (p1 mod 10^(k-1))
|
|
// = (d[k-1] ) * 10^(k-1) + (p1 mod 10^(k-1))
|
|
//
|
|
// M+ = p1 + p2 * 2^e
|
|
// = d[k-1] * 10^(k-1) + (p1 mod 10^(k-1)) + p2 * 2^e
|
|
// = d[k-1] * 10^(k-1) + ((p1 mod 10^(k-1)) * 2^-e + p2) * 2^e
|
|
// = d[k-1] * 10^(k-1) + ( rest) * 2^e
|
|
//
|
|
// Now generate the digits d[n] of p1 from left to right (n = k-1,...,0)
|
|
//
|
|
// p1 = d[k-1]...d[n] * 10^n + d[n-1]...d[0]
|
|
//
|
|
// but stop as soon as
|
|
//
|
|
// rest * 2^e = (d[n-1]...d[0] * 2^-e + p2) * 2^e <= delta * 2^e
|
|
|
|
int n = k;
|
|
while (n > 0)
|
|
{
|
|
// Invariants:
|
|
// M+ = buffer * 10^n + (p1 + p2 * 2^e) (buffer = 0 for n = k)
|
|
// pow10 = 10^(n-1) <= p1 < 10^n
|
|
//
|
|
const uint32_t d = p1 / pow10; // d = p1 div 10^(n-1)
|
|
const uint32_t r = p1 % pow10; // r = p1 mod 10^(n-1)
|
|
//
|
|
// M+ = buffer * 10^n + (d * 10^(n-1) + r) + p2 * 2^e
|
|
// = (buffer * 10 + d) * 10^(n-1) + (r + p2 * 2^e)
|
|
//
|
|
assert(d <= 9);
|
|
buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
|
|
//
|
|
// M+ = buffer * 10^(n-1) + (r + p2 * 2^e)
|
|
//
|
|
p1 = r;
|
|
n--;
|
|
//
|
|
// M+ = buffer * 10^n + (p1 + p2 * 2^e)
|
|
// pow10 = 10^n
|
|
//
|
|
|
|
// Now check if enough digits have been generated.
|
|
// Compute
|
|
//
|
|
// p1 + p2 * 2^e = (p1 * 2^-e + p2) * 2^e = rest * 2^e
|
|
//
|
|
// Note:
|
|
// Since rest and delta share the same exponent e, it suffices to
|
|
// compare the significands.
|
|
const uint64_t rest = (uint64_t{p1} << -one.e) + p2;
|
|
if (rest <= delta)
|
|
{
|
|
// V = buffer * 10^n, with M- <= V <= M+.
|
|
|
|
decimal_exponent += n;
|
|
|
|
// We may now just stop. But instead look if the buffer could be
|
|
// decremented to bring V closer to w.
|
|
//
|
|
// pow10 = 10^n is now 1 ulp in the decimal representation V.
|
|
// The rounding procedure works with diyfp's with an implicit
|
|
// exponent of e.
|
|
//
|
|
// 10^n = (10^n * 2^-e) * 2^e = ulp * 2^e
|
|
//
|
|
const uint64_t ten_n = uint64_t{pow10} << -one.e;
|
|
grisu2_round(buffer, length, dist, delta, rest, ten_n);
|
|
|
|
return;
|
|
}
|
|
|
|
pow10 /= 10;
|
|
//
|
|
// pow10 = 10^(n-1) <= p1 < 10^n
|
|
// Invariants restored.
|
|
}
|
|
|
|
// 2)
|
|
//
|
|
// The digits of the integral part have been generated:
|
|
//
|
|
// M+ = d[k-1]...d[1]d[0] + p2 * 2^e
|
|
// = buffer + p2 * 2^e
|
|
//
|
|
// Now generate the digits of the fractional part p2 * 2^e.
|
|
//
|
|
// Note:
|
|
// No decimal point is generated: the exponent is adjusted instead.
|
|
//
|
|
// p2 actually represents the fraction
|
|
//
|
|
// p2 * 2^e
|
|
// = p2 / 2^-e
|
|
// = d[-1] / 10^1 + d[-2] / 10^2 + ...
|
|
//
|
|
// Now generate the digits d[-m] of p1 from left to right (m = 1,2,...)
|
|
//
|
|
// p2 * 2^e = d[-1]d[-2]...d[-m] * 10^-m
|
|
// + 10^-m * (d[-m-1] / 10^1 + d[-m-2] / 10^2 + ...)
|
|
//
|
|
// using
|
|
//
|
|
// 10^m * p2 = ((10^m * p2) div 2^-e) * 2^-e + ((10^m * p2) mod 2^-e)
|
|
// = ( d) * 2^-e + ( r)
|
|
//
|
|
// or
|
|
// 10^m * p2 * 2^e = d + r * 2^e
|
|
//
|
|
// i.e.
|
|
//
|
|
// M+ = buffer + p2 * 2^e
|
|
// = buffer + 10^-m * (d + r * 2^e)
|
|
// = (buffer * 10^m + d) * 10^-m + 10^-m * r * 2^e
|
|
//
|
|
// and stop as soon as 10^-m * r * 2^e <= delta * 2^e
|
|
|
|
assert(p2 > delta);
|
|
|
|
int m = 0;
|
|
for (;;)
|
|
{
|
|
// Invariant:
|
|
// M+ = buffer * 10^-m + 10^-m * (d[-m-1] / 10 + d[-m-2] / 10^2 + ...) * 2^e
|
|
// = buffer * 10^-m + 10^-m * (p2 ) * 2^e
|
|
// = buffer * 10^-m + 10^-m * (1/10 * (10 * p2) ) * 2^e
|
|
// = buffer * 10^-m + 10^-m * (1/10 * ((10*p2 div 2^-e) * 2^-e + (10*p2 mod 2^-e)) * 2^e
|
|
//
|
|
assert(p2 <= UINT64_MAX / 10);
|
|
p2 *= 10;
|
|
const uint64_t d = p2 >> -one.e; // d = (10 * p2) div 2^-e
|
|
const uint64_t r = p2 & (one.f - 1); // r = (10 * p2) mod 2^-e
|
|
//
|
|
// M+ = buffer * 10^-m + 10^-m * (1/10 * (d * 2^-e + r) * 2^e
|
|
// = buffer * 10^-m + 10^-m * (1/10 * (d + r * 2^e))
|
|
// = (buffer * 10 + d) * 10^(-m-1) + 10^(-m-1) * r * 2^e
|
|
//
|
|
assert(d <= 9);
|
|
buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
|
|
//
|
|
// M+ = buffer * 10^(-m-1) + 10^(-m-1) * r * 2^e
|
|
//
|
|
p2 = r;
|
|
m++;
|
|
//
|
|
// M+ = buffer * 10^-m + 10^-m * p2 * 2^e
|
|
// Invariant restored.
|
|
|
|
// Check if enough digits have been generated.
|
|
//
|
|
// 10^-m * p2 * 2^e <= delta * 2^e
|
|
// p2 * 2^e <= 10^m * delta * 2^e
|
|
// p2 <= 10^m * delta
|
|
delta *= 10;
|
|
dist *= 10;
|
|
if (p2 <= delta)
|
|
{
|
|
break;
|
|
}
|
|
}
|
|
|
|
// V = buffer * 10^-m, with M- <= V <= M+.
|
|
|
|
decimal_exponent -= m;
|
|
|
|
// 1 ulp in the decimal representation is now 10^-m.
|
|
// Since delta and dist are now scaled by 10^m, we need to do the
|
|
// same with ulp in order to keep the units in sync.
|
|
//
|
|
// 10^m * 10^-m = 1 = 2^-e * 2^e = ten_m * 2^e
|
|
//
|
|
const uint64_t ten_m = one.f;
|
|
grisu2_round(buffer, length, dist, delta, p2, ten_m);
|
|
|
|
// By construction this algorithm generates the shortest possible decimal
|
|
// number (Loitsch, Theorem 6.2) which rounds back to w.
|
|
// For an input number of precision p, at least
|
|
//
|
|
// N = 1 + ceil(p * log_10(2))
|
|
//
|
|
// decimal digits are sufficient to identify all binary floating-point
|
|
// numbers (Matula, "In-and-Out conversions").
|
|
// This implies that the algorithm does not produce more than N decimal
|
|
// digits.
|
|
//
|
|
// N = 17 for p = 53 (IEEE double precision)
|
|
// N = 9 for p = 24 (IEEE single precision)
|
|
}
|
|
|
|
// v = buf * 10^decimal_exponent
|
|
// len is the length of the buffer (number of decimal digits)
|
|
// The buffer must be large enough, i.e. >= max_digits10.
|
|
inline void grisu2(char* buf, int& len, int& decimal_exponent, diyfp m_minus, diyfp v, diyfp m_plus)
|
|
{
|
|
assert(m_plus.e == m_minus.e);
|
|
assert(m_plus.e == v.e);
|
|
|
|
// --------(-----------------------+-----------------------)-------- (A)
|
|
// m- v m+
|
|
//
|
|
// --------------------(-----------+-----------------------)-------- (B)
|
|
// m- v m+
|
|
//
|
|
// First scale v (and m- and m+) such that the exponent is in the range
|
|
// [alpha, gamma].
|
|
|
|
const cached_power cached = get_cached_power_for_binary_exponent(m_plus.e);
|
|
|
|
const diyfp c_minus_k(cached.f, cached.e); // = c ~= 10^-k
|
|
|
|
// The exponent of the products is = v.e + c_minus_k.e + q and is in the range [alpha,gamma]
|
|
const diyfp w = diyfp::mul(v, c_minus_k);
|
|
const diyfp w_minus = diyfp::mul(m_minus, c_minus_k);
|
|
const diyfp w_plus = diyfp::mul(m_plus, c_minus_k);
|
|
|
|
// ----(---+---)---------------(---+---)---------------(---+---)----
|
|
// w- w w+
|
|
// = c*m- = c*v = c*m+
|
|
//
|
|
// diyfp::mul rounds its result and c_minus_k is approximated too. w, w- and
|
|
// w+ are now off by a small amount.
|
|
// In fact:
|
|
//
|
|
// w - v * 10^k < 1 ulp
|
|
//
|
|
// To account for this inaccuracy, add resp. subtract 1 ulp.
|
|
//
|
|
// --------+---[---------------(---+---)---------------]---+--------
|
|
// w- M- w M+ w+
|
|
//
|
|
// Now any number in [M-, M+] (bounds included) will round to w when input,
|
|
// regardless of how the input rounding algorithm breaks ties.
|
|
//
|
|
// And digit_gen generates the shortest possible such number in [M-, M+].
|
|
// Note that this does not mean that Grisu2 always generates the shortest
|
|
// possible number in the interval (m-, m+).
|
|
const diyfp M_minus(w_minus.f + 1, w_minus.e);
|
|
const diyfp M_plus (w_plus.f - 1, w_plus.e );
|
|
|
|
decimal_exponent = -cached.k; // = -(-k) = k
|
|
|
|
grisu2_digit_gen(buf, len, decimal_exponent, M_minus, w, M_plus);
|
|
}
|
|
|
|
// v = buf * 10^decimal_exponent
|
|
// len is the length of the buffer (number of decimal digits)
|
|
// The buffer must be large enough, i.e. >= max_digits10.
|
|
template <typename FloatType>
|
|
void grisu2(char* buf, int& len, int& decimal_exponent, FloatType value)
|
|
{
|
|
static_assert(diyfp::kPrecision >= std::numeric_limits<FloatType>::digits + 3,
|
|
"internal error: not enough precision");
|
|
|
|
assert(std::isfinite(value));
|
|
assert(value > 0);
|
|
|
|
// If the neighbors (and boundaries) of 'value' are always computed for double-precision
|
|
// numbers, all float's can be recovered using strtod (and strtof). However, the resulting
|
|
// decimal representations are not exactly "short".
|
|
//
|
|
// The documentation for 'std::to_chars' (http://en.cppreference.com/w/cpp/utility/to_chars)
|
|
// says "value is converted to a string as if by std::sprintf in the default ("C") locale"
|
|
// and since sprintf promotes float's to double's, I think this is exactly what 'std::to_chars'
|
|
// does.
|
|
// On the other hand, the documentation for 'std::to_chars' requires that "parsing the
|
|
// representation using the corresponding std::from_chars function recovers value exactly". That
|
|
// indicates that single precision floating-point numbers should be recovered using
|
|
// 'std::strtof'.
|
|
//
|
|
// NB: If the neighbors are computed for single-precision numbers, there is a single float
|
|
// (7.0385307e-26f) which can't be recovered using strtod. The resulting double precision
|
|
// value is off by 1 ulp.
|
|
#if 0
|
|
const boundaries w = compute_boundaries(static_cast<double>(value));
|
|
#else
|
|
const boundaries w = compute_boundaries(value);
|
|
#endif
|
|
|
|
grisu2(buf, len, decimal_exponent, w.minus, w.w, w.plus);
|
|
}
|
|
|
|
// Appends a decimal representation of e to buf.
|
|
// Returns a pointer to the element following the exponent.
|
|
// PRE: -1000 < e < 1000
|
|
inline char* append_exponent(char* buf, int e)
|
|
{
|
|
assert(e > -1000);
|
|
assert(e < 1000);
|
|
|
|
if (e < 0)
|
|
{
|
|
e = -e;
|
|
*buf++ = '-';
|
|
}
|
|
else
|
|
{
|
|
*buf++ = '+';
|
|
}
|
|
|
|
uint32_t k = static_cast<uint32_t>(e);
|
|
if (k < 10)
|
|
{
|
|
// Always print at least two digits in the exponent.
|
|
// This is for compatibility with printf("%g").
|
|
*buf++ = '0';
|
|
*buf++ = static_cast<char>('0' + k);
|
|
}
|
|
else if (k < 100)
|
|
{
|
|
*buf++ = static_cast<char>('0' + k / 10);
|
|
k %= 10;
|
|
*buf++ = static_cast<char>('0' + k);
|
|
}
|
|
else
|
|
{
|
|
*buf++ = static_cast<char>('0' + k / 100);
|
|
k %= 100;
|
|
*buf++ = static_cast<char>('0' + k / 10);
|
|
k %= 10;
|
|
*buf++ = static_cast<char>('0' + k);
|
|
}
|
|
|
|
return buf;
|
|
}
|
|
|
|
// Prettify v = buf * 10^decimal_exponent
|
|
// If v is in the range [10^min_exp, 10^max_exp) it will be printed in fixed-point notation.
|
|
// Otherwise it will be printed in exponential notation.
|
|
// PRE: min_exp < 0
|
|
// PRE: max_exp > 0
|
|
inline char* format_buffer(char* buf, int len, int decimal_exponent, int min_exp, int max_exp)
|
|
{
|
|
assert(min_exp < 0);
|
|
assert(max_exp > 0);
|
|
|
|
const int k = len;
|
|
const int n = len + decimal_exponent;
|
|
|
|
// v = buf * 10^(n-k)
|
|
// k is the length of the buffer (number of decimal digits)
|
|
// n is the position of the decimal point relative to the start of the buffer.
|
|
|
|
if (k <= n and n <= max_exp)
|
|
{
|
|
// digits[000]
|
|
// len <= max_exp + 2
|
|
|
|
std::memset(buf + k, '0', static_cast<size_t>(n - k));
|
|
// Make it look like a floating-point number (#362, #378)
|
|
buf[n + 0] = '.';
|
|
buf[n + 1] = '0';
|
|
return buf + (n + 2);
|
|
}
|
|
|
|
if (0 < n and n <= max_exp)
|
|
{
|
|
// dig.its
|
|
// len <= max_digits10 + 1
|
|
|
|
assert(k > n);
|
|
|
|
std::memmove(buf + (n + 1), buf + n, static_cast<size_t>(k - n));
|
|
buf[n] = '.';
|
|
return buf + (k + 1);
|
|
}
|
|
|
|
if (min_exp < n and n <= 0)
|
|
{
|
|
// 0.[000]digits
|
|
// len <= 2 + (-min_exp - 1) + max_digits10
|
|
|
|
std::memmove(buf + (2 + -n), buf, static_cast<size_t>(k));
|
|
buf[0] = '0';
|
|
buf[1] = '.';
|
|
std::memset(buf + 2, '0', static_cast<size_t>(-n));
|
|
return buf + (2 + (-n) + k);
|
|
}
|
|
|
|
if (k == 1)
|
|
{
|
|
// dE+123
|
|
// len <= 1 + 5
|
|
|
|
buf += 1;
|
|
}
|
|
else
|
|
{
|
|
// d.igitsE+123
|
|
// len <= max_digits10 + 1 + 5
|
|
|
|
std::memmove(buf + 2, buf + 1, static_cast<size_t>(k - 1));
|
|
buf[1] = '.';
|
|
buf += 1 + k;
|
|
}
|
|
|
|
*buf++ = 'e';
|
|
return append_exponent(buf, n - 1);
|
|
}
|
|
|
|
} // namespace dtoa_impl
|
|
|
|
// Generates a decimal representation of the floating-point number value in [first, last).
|
|
//
|
|
// The format of the resulting decimal representation is similar to printf's %g format.
|
|
// Returns an iterator pointing past-the-end of the decimal representation.
|
|
//
|
|
// Note: The input number must be finite, i.e. NaN's and Inf's are not supported.
|
|
// Note: The buffer must be large enough.
|
|
// Note: The result is NOT null-terminated.
|
|
template <typename FloatType>
|
|
char* to_chars(char* first, char* last, FloatType value)
|
|
{
|
|
static_cast<void>(last); // maybe unused - fix warning
|
|
assert(std::isfinite(value));
|
|
|
|
// Use signbit(value) instead of (value < 0) since signbit works for -0.
|
|
if (std::signbit(value))
|
|
{
|
|
value = -value;
|
|
*first++ = '-';
|
|
}
|
|
|
|
if (value == 0) // +-0
|
|
{
|
|
*first++ = '0';
|
|
// Make it look like a floating-point number (#362, #378)
|
|
*first++ = '.';
|
|
*first++ = '0';
|
|
return first;
|
|
}
|
|
|
|
assert(last - first >= std::numeric_limits<FloatType>::max_digits10);
|
|
|
|
// Compute v = buffer * 10^decimal_exponent.
|
|
// The decimal digits are stored in the buffer, which needs to be interpreted
|
|
// as an unsigned decimal integer.
|
|
// len is the length of the buffer, i.e. the number of decimal digits.
|
|
int len = 0;
|
|
int decimal_exponent = 0;
|
|
dtoa_impl::grisu2(first, len, decimal_exponent, value);
|
|
|
|
assert(len <= std::numeric_limits<FloatType>::max_digits10);
|
|
|
|
// Format the buffer like printf("%.*g", prec, value)
|
|
constexpr int kMinExp = -4;
|
|
// Use digits10 here to increase compatibility with version 2.
|
|
constexpr int kMaxExp = std::numeric_limits<FloatType>::digits10;
|
|
|
|
assert(last - first >= kMaxExp + 2);
|
|
assert(last - first >= 2 + (-kMinExp - 1) + std::numeric_limits<FloatType>::max_digits10);
|
|
assert(last - first >= std::numeric_limits<FloatType>::max_digits10 + 6);
|
|
|
|
return dtoa_impl::format_buffer(first, len, decimal_exponent, kMinExp, kMaxExp);
|
|
}
|
|
|
|
} // namespace detail
|
|
} // namespace nlohmann
|