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/*
* Copyright 2018-2020 Raffaello Giulietti
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
package com.fasterxml.jackson.core.io.schubfach;
import static com.fasterxml.jackson.core.io.schubfach.MathUtils.flog10pow2;
import static com.fasterxml.jackson.core.io.schubfach.MathUtils.flog10threeQuartersPow2;
import static com.fasterxml.jackson.core.io.schubfach.MathUtils.flog2pow10;
import static com.fasterxml.jackson.core.io.schubfach.MathUtils.g1;
import static com.fasterxml.jackson.core.io.schubfach.MathUtils.multiplyHigh;
import static com.fasterxml.jackson.core.io.schubfach.MathUtils.pow10;
import static java.lang.Float.floatToRawIntBits;
import static java.lang.Integer.numberOfLeadingZeros;
/**
* This class exposes a method to render a {@code float} as a string.
*
* @author Raffaello Giulietti
*/
final public class FloatToDecimal {
/*
For full details about this code see the following references:
[1] Giulietti, "The Schubfach way to render doubles",
https://drive.google.com/open?id=1luHhyQF9zKlM8yJ1nebU0OgVYhfC6CBN
[2] IEEE Computer Society, "IEEE Standard for Floating-Point Arithmetic"
[3] Bouvier & Zimmermann, "Division-Free Binary-to-Decimal Conversion"
Divisions are avoided altogether for the benefit of those architectures
that do not provide specific machine instructions or where they are slow.
This is discussed in section 10 of [1].
*/
// Sources with the license are here: https://github.com/c4f7fcce9cb06515/Schubfach/blob/3c92d3c9b1fead540616c918cdfef432bca53dfa/todec/src/math/FloatToDecimal.java
// The precision in bits.
static final int P = 24;
// Exponent width in bits.
private static final int W = (Float.SIZE - 1) - (P - 1);
// Minimum value of the exponent: -(2^(W-1)) - P + 3.
static final int Q_MIN = (-1 << W - 1) - P + 3;
// Maximum value of the exponent: 2^(W-1) - P.
static final int Q_MAX = (1 << W - 1) - P;
// 10^(E_MIN - 1) <= MIN_VALUE < 10^E_MIN
static final int E_MIN = -44;
// 10^(E_MAX - 1) <= MAX_VALUE < 10^E_MAX
static final int E_MAX = 39;
// Threshold to detect tiny values, as in section 8.1.1 of [1]
static final int C_TINY = 8;
// The minimum and maximum k, as in section 8 of [1]
static final int K_MIN = -45;
static final int K_MAX = 31;
// H is as in section 8 of [1].
static final int H = 9;
// Minimum value of the significand of a normal value: 2^(P-1).
private static final int C_MIN = 1 << P - 1;
// Mask to extract the biased exponent.
private static final int BQ_MASK = (1 << W) - 1;
// Mask to extract the fraction bits.
private static final int T_MASK = (1 << P - 1) - 1;
// Used in rop().
private static final long MASK_32 = (1L << 32) - 1;
// Used for left-to-tight digit extraction.
private static final int MASK_28 = (1 << 28) - 1;
private static final int NON_SPECIAL = 0;
private static final int PLUS_ZERO = 1;
private static final int MINUS_ZERO = 2;
private static final int PLUS_INF = 3;
private static final int MINUS_INF = 4;
private static final int NAN = 5;
/*
Room for the longer of the forms
-ddddd.dddd H + 2 characters
-0.00ddddddddd H + 5 characters
-d.ddddddddE-ee H + 6 characters
where there are H digits d
*/
public final int MAX_CHARS = H + 6;
// Numerical results are created here...
private final byte[] bytes = new byte[MAX_CHARS];
// Index into buf of rightmost valid character.
private int index;
private FloatToDecimal() {
}
/**
* Returns a string rendering of the {@code float} argument.
*
* The characters of the result are all drawn from the ASCII set.
*
* - Any NaN, whether quiet or signaling, is rendered as
* {@code "NaN"}, regardless of the sign bit.
*
- The infinities +∞ and -∞ are rendered as
* {@code "Infinity"} and {@code "-Infinity"}, respectively.
*
- The positive and negative zeroes are rendered as
* {@code "0.0"} and {@code "-0.0"}, respectively.
*
- A finite negative {@code v} is rendered as the sign
* '{@code -}' followed by the rendering of the magnitude -{@code v}.
*
- A finite positive {@code v} is rendered in two stages:
*
* - Selection of a decimal: A well-defined
* decimal d
v is selected
* to represent {@code v}.
* - Formatting as a string: The decimal
* d
v is formatted as a string,
* either in plain or in computerized scientific notation,
* depending on its value.
*
*
*
* A decimal is a number of the form
* d×10i
* for some (unique) integers d > 0 and i such that
* d is not a multiple of 10.
* These integers are the significand and
* the exponent, respectively, of the decimal.
* The length of the decimal is the (unique)
* integer n meeting
* 10n-1 ≤ d < 10n.
*
*
The decimal dv
* for a finite positive {@code v} is defined as follows:
*
* - Let R be the set of all decimals that round to {@code v}
* according to the usual round-to-closest rule of
* IEEE 754 floating-point arithmetic.
*
- Let m be the minimal length over all decimals in R.
*
- When m ≥ 2, let T be the set of all decimals
* in R with length m.
* Otherwise, let T be the set of all decimals
* in R with length 1 or 2.
*
- Define d
v as
* the decimal in T that is closest to {@code v}.
* Or if there are two such decimals in T,
* select the one with the even significand (there is exactly one).
*
*
* The (uniquely) selected decimal dv
* is then formatted.
*
*
Let d, i and n be the significand, exponent and
* length of dv, respectively.
* Further, let e = n + i - 1 and let
* d1…dn
* be the usual decimal expansion of the significand.
* Note that d1 ≠ 0 ≠ dn.
*
* - Case -3 ≤ e < 0:
* d
v is formatted as
* 0.0…0d1…dn,
* where there are exactly -(n + i) zeroes between
* the decimal point and d1.
* For example, 123 × 10-4 is formatted as
* {@code 0.0123}.
* - Case 0 ≤ e < 7:
*
* - Subcase i ≥ 0:
* d
v is formatted as
* d1…dn0…0.0,
* where there are exactly i zeroes
* between dn and the decimal point.
* For example, 123 × 102 is formatted as
* {@code 12300.0}.
* - Subcase i < 0:
* d
v is formatted as
* d1…dn+i.dn+i+1…dn.
* There are exactly -i digits to the right of
* the decimal point.
* For example, 123 × 10-1 is formatted as
* {@code 12.3}.
*
* - Case e < -3 or e ≥ 7:
* computerized scientific notation is used to format
* d
v.
* Here e is formatted as by {@link Integer#toString(int)}.
*
* - Subcase n = 1:
* d
v is formatted as
* d1.0Ee.
* For example, 1 × 1023 is formatted as
* {@code 1.0E23}.
* - Subcase n > 1:
* d
v is formatted as
* d1.d2…dnEe.
* For example, 123 × 10-21 is formatted as
* {@code 1.23E-19}.
*
*
*
* @param v the {@code float} to be rendered.
* @return a string rendering of the argument.
*/
public static String toString(float v) {
return new FloatToDecimal().toDecimalString(v);
}
private String toDecimalString(float v) {
switch (toDecimal(v)) {
case NON_SPECIAL: return charsToString();
case PLUS_ZERO: return "0.0";
case MINUS_ZERO: return "-0.0";
case PLUS_INF: return "Infinity";
case MINUS_INF: return "-Infinity";
default: return "NaN";
}
}
/*
Returns
PLUS_ZERO iff v is 0.0
MINUS_ZERO iff v is -0.0
PLUS_INF iff v is POSITIVE_INFINITY
MINUS_INF iff v is NEGATIVE_INFINITY
NAN iff v is NaN
*/
private int toDecimal(float v) {
/*
For full details see references [2] and [1].
For finite v != 0, determine integers c and q such that
|v| = c 2^q and
Q_MIN <= q <= Q_MAX and
either 2^(P-1) <= c < 2^P (normal)
or 0 < c < 2^(P-1) and q = Q_MIN (subnormal)
*/
int bits = floatToRawIntBits(v);
int t = bits & T_MASK;
int bq = (bits >>> P - 1) & BQ_MASK;
if (bq < BQ_MASK) {
index = -1;
if (bits < 0) {
append('-');
}
if (bq != 0) {
// normal value. Here mq = -q
int mq = -Q_MIN + 1 - bq;
int c = C_MIN | t;
// The fast path discussed in section 8.2 of [1].
if (0 < mq & mq < P) {
int f = c >> mq;
if (f << mq == c) {
return toChars(f, 0);
}
}
return toDecimal(-mq, c, 0);
}
if (t != 0) {
// subnormal value
return t < C_TINY
? toDecimal(Q_MIN, 10 * t, -1)
: toDecimal(Q_MIN, t, 0);
}
return bits == 0 ? PLUS_ZERO : MINUS_ZERO;
}
if (t != 0) {
return NAN;
}
return bits > 0 ? PLUS_INF : MINUS_INF;
}
private int toDecimal(int q, int c, int dk) {
/*
The skeleton corresponds to figure 4 of [1].
The efficient computations are those summarized in figure 7.
Also check the appendix.
Here's a correspondence between Java names and names in [1],
expressed as approximate LaTeX source code and informally.
Other names are identical.
cb: \bar{c} "c-bar"
cbr: \bar{c}_r "c-bar-r"
cbl: \bar{c}_l "c-bar-l"
vb: \bar{v} "v-bar"
vbr: \bar{v}_r "v-bar-r"
vbl: \bar{v}_l "v-bar-l"
rop: r_o' "r-o-prime"
*/
int out = c & 0x1;
long cb = c << 2;
long cbr = cb + 2;
long cbl;
int k;
/*
flog10pow2(e) = floor(log_10(2^e))
flog10threeQuartersPow2(e) = floor(log_10(3/4 2^e))
flog2pow10(e) = floor(log_2(10^e))
*/
if (c != C_MIN | q == Q_MIN) {
// regular spacing
cbl = cb - 2;
k = flog10pow2(q);
} else {
// irregular spacing0
cbl = cb - 1;
k = flog10threeQuartersPow2(q);
}
int h = q + flog2pow10(-k) + 33;
// g is as in the appendix
long g = g1(k) + 1;
int vb = rop(g, cb << h);
int vbl = rop(g, cbl << h);
int vbr = rop(g, cbr << h);
int s = vb >> 2;
if (s >= 100) {
/*
For n = 9, m = 1 the table in section 10 of [1] shows
s' = floor(s / 10) = floor(s 1_717_986_919 / 2^34)
sp10 = 10 s'
tp10 = 10 t'
upin iff u' = sp10 10^k in Rv
wpin iff w' = tp10 10^k in Rv
See section 9.4 of [1].
*/
int sp10 = 10 * (int) (s * 1_717_986_919L >>> 34);
int tp10 = sp10 + 10;
boolean upin = vbl + out <= sp10 << 2;
boolean wpin = (tp10 << 2) + out <= vbr;
if (upin != wpin) {
return toChars(upin ? sp10 : tp10, k);
}
}
/*
10 <= s < 100 or s >= 100 and u', w' not in Rv
uin iff u = s 10^k in Rv
win iff w = t 10^k in Rv
See section 9.4 of [1].
*/
int t = s + 1;
boolean uin = vbl + out <= s << 2;
boolean win = (t << 2) + out <= vbr;
if (uin != win) {
// Exactly one of u or w lies in Rv.
return toChars(uin ? s : t, k + dk);
}
/*
Both u and w lie in Rv: determine the one closest to v.
See section 9.4 of [1].
*/
int cmp = vb - (s + t << 1);
return toChars(cmp < 0 || cmp == 0 && (s & 0x1) == 0 ? s : t, k + dk);
}
/*
Computes rop(cp g 2^(-95))
See appendix and figure 8 of [1].
*/
private static int rop(long g, long cp) {
long x1 = multiplyHigh(g, cp);
long vbp = x1 >>> 31;
return (int) (vbp | (x1 & MASK_32) + MASK_32 >>> 32);
}
/*
Formats the decimal f 10^e.
*/
private int toChars(int f, int e) {
/*
For details not discussed here see section 10 of [1].
Determine len such that
10^(len-1) <= f < 10^len
*/
int len = flog10pow2(Integer.SIZE - numberOfLeadingZeros(f));
if (f >= pow10(len)) {
len += 1;
}
/*
Let fp and ep be the original f and e, respectively.
Transform f and e to ensure
10^(H-1) <= f < 10^H
fp 10^ep = f 10^(e-H) = 0.f 10^e
*/
f *= pow10(H - len);
e += len;
/*
The toChars?() methods perform left-to-right digits extraction
using ints, provided that the arguments are limited to 8 digits.
Therefore, split the H = 9 digits of f into:
h = the most significant digit of f
l = the last 8, least significant digits of f
For n = 9, m = 8 the table in section 10 of [1] shows
floor(f / 10^8) = floor(1_441_151_881 f / 2^57)
*/
int h = (int) (f * 1_441_151_881L >>> 57);
int l = f - 100_000_000 * h;
if (0 < e && e <= 7) {
return toChars1(h, l, e);
}
if (-3 < e && e <= 0) {
return toChars2(h, l, e);
}
return toChars3(h, l, e);
}
private int toChars1(int h, int l, int e) {
/*
0 < e <= 7: plain format without leading zeroes.
Left-to-right digits extraction:
algorithm 1 in [3], with b = 10, k = 8, n = 28.
*/
appendDigit(h);
int y = y(l);
int t;
int i = 1;
for (; i < e; ++i) {
t = 10 * y;
appendDigit(t >>> 28);
y = t & MASK_28;
}
append('.');
for (; i <= 8; ++i) {
t = 10 * y;
appendDigit(t >>> 28);
y = t & MASK_28;
}
removeTrailingZeroes();
return NON_SPECIAL;
}
private int toChars2(int h, int l, int e) {
// -3 < e <= 0: plain format with leading zeroes.
appendDigit(0);
append('.');
for (; e < 0; ++e) {
appendDigit(0);
}
appendDigit(h);
append8Digits(l);
removeTrailingZeroes();
return NON_SPECIAL;
}
private int toChars3(int h, int l, int e) {
// -3 >= e | e > 7: computerized scientific notation
appendDigit(h);
append('.');
append8Digits(l);
removeTrailingZeroes();
exponent(e - 1);
return NON_SPECIAL;
}
private void append8Digits(int m) {
/*
Left-to-right digits extraction:
algorithm 1 in [3], with b = 10, k = 8, n = 28.
*/
int y = y(m);
for (int i = 0; i < 8; ++i) {
int t = 10 * y;
appendDigit(t >>> 28);
y = t & MASK_28;
}
}
private void removeTrailingZeroes() {
while (bytes[index] == '0') {
--index;
}
// ... but do not remove the one directly to the right of '.'
if (bytes[index] == '.') {
++index;
}
}
private int y(int a) {
/*
Algorithm 1 in [3] needs computation of
floor((a + 1) 2^n / b^k) - 1
with a < 10^8, b = 10, k = 8, n = 28.
Noting that
(a + 1) 2^n <= 10^8 2^28 < 10^17
For n = 17, m = 8 the table in section 10 of [1] leads to:
*/
return (int) (multiplyHigh(
(long) (a + 1) << 28,
193_428_131_138_340_668L) >>> 20) - 1;
}
private void exponent(int e) {
append('E');
if (e < 0) {
append('-');
e = -e;
}
if (e < 10) {
appendDigit(e);
return;
}
/*
For n = 2, m = 1 the table in section 10 of [1] shows
floor(e / 10) = floor(103 e / 2^10)
*/
int d = e * 103 >>> 10;
appendDigit(d);
appendDigit(e - 10 * d);
}
private void append(int c) {
bytes[++index] = (byte) c;
}
private void appendDigit(int d) {
bytes[++index] = (byte) ('0' + d);
}
// Using the deprecated constructor enhances performance.
@SuppressWarnings("deprecation")
private String charsToString() {
return new String(bytes, 0, 0, index + 1);
}
}
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