Java Hangs When Converting 2.2250738585072012e-308

Konstantin Preißer made an interesting discovery, after reading my article “PHP Hangs On Numeric Value 2.2250738585072011e-308”: Java — both its runtime and compiler — go into an infinite loop when converting the decimal number 2.2250738585072012e-308 to double-precision binary floating-point. This number is supposed to convert to 0x1p-1022, which is DBL_MIN; instead, Java gets stuck, oscillating between 0x1p-1022 and 0x0.fffffffffffffp-1022, the largest subnormal double-precision floating-point number.

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Why “Volatile” Fixes the 2.2250738585072011e-308 Bug

Recently I discovered a serious bug in x87 builds of PHP: PHP’s decimal to floating-point conversion routine, zend_strtod(), went into an infinite loop when converting the decimal string 2.2250738585072011e-308 to double-precision binary floating-point. This problem was fixed with a simple one line of code change to zend_strtod.c:

This line

double aadj, aadj1, adj;

was changed to

volatile double aadj, aadj1, adj;

Why does this fix the problem? I uncovered the very specific reason: it prevents a double rounding on underflow error.

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PHP Hangs On Numeric Value 2.2250738585072011e-308

I stumbled upon a very strange bug in PHP; this statement sends it into an infinite loop:

<?php $d = 2.2250738585072011e-308; ?>

(The same thing happens if you write the number without scientific notation — 324 decimal places.)

I hit this bug in the two places I tested for it: on Windows (PHP 5.3.1 under XAMPP 1.7.3), and on Linux (PHP Version 5.3.2-1ubuntu4.5) — both on an Intel Core Duo processor. I’ve written a bug report.

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Fifteen Digits Don’t Round-Trip Through SQLite Reals

I’ve discovered that decimal floating-point numbers of 15 significant digits or less don’t always round-trip through SQLite. Consider this example, executed on version 3.7.3 of the pre-compiled SQLite command shell:

sqlite> create table t1(d real);
sqlite> insert into t1 values(9.944932e+31);
sqlite> select * from t1;
9.94493200000001e+31

SQLite represents a decimal floating-point number that has real affinity as a double-precision binary floating-point number — a double. A decimal number of 15 significant digits or less is supposed to be recoverable from its double-precision representation. In SQLite, however, this guarantee is not met; this is because its floating-point to decimal conversion routine is implemented in limited-precision floating-point arithmetic.

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The Answer is One (Unless You Use Floating-Point)

What does this C function do?

double f(double a)
{
 double b, c;

 b = 10*a - 10;
 c = a - 0.1*b;

 return (c);
}

Based solely on reading the code, you’ll conclude that it always returns 1: c = a – 0.1*(10*a – 10) = a – (a-1) = 1. But if you execute the code, you’ll find that it may or may not return 1, depending on the input. If you know anything about binary floating-point arithmetic, that won’t surprise you; what might surprise you is how far from 1 the answer can be — as far away as a large negative number!

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Quick and Dirty Floating-Point to Decimal Conversion

In my article “Quick and Dirty Decimal to Floating-Point Conversion” I presented a small C program that uses double-precision floating-point arithmetic to convert decimal strings to binary floating-point numbers. The program converts some numbers incorrectly, despite using an algorithm that’s mathematically correct; its limited precision calculations are to blame. I dubbed the program “quick and dirty” because it’s simple, and overall converts reasonably accurately.

For this article, I took a similar approach to the conversion in the opposite direction — from binary floating-point to decimal string. I wrote a small C program that combines two mathematically correct algorithms: the classic “repeated division by ten” algorithm to convert integer values, and the classic “repeated multiplication by ten” algorithm to convert fractional values. The program uses double-precision floating-point arithmetic, so like its quick and dirty decimal to floating-point counterpart, its conversions are not always correct — though reasonably accurate. I’ll present the program and analyze some example conversions, both correct and incorrect.

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Inconsistent Rounding of Printed Floating-Point Numbers

What does this C program print?

#include <stdio.h>
int main (void)
{
 printf ("%.1f\n",0.25);
}

The answer depends on which compiler you use. If you compile the program with Visual C++ and run on it on Windows, it prints 0.3; if you compile it with gcc and run it on Linux, it prints 0.2.

The compilers — actually, their run time libraries — are using different rules to break decimal rounding ties. The two-digit number 0.25, which has an exact binary floating-point representation, is equally near two one-digit decimal numbers: 0.2 and 0.3; either is an acceptable answer. Visual C++ uses the round-half-away-from-zero rule, and gcc (actually, glibc) uses the round-half-to-even rule, also known as bankers’ rounding.

This inconsistency of printed output is not limited to C — it spans many programming environments. In all, I tested fixed-format printing in nineteen environments: in thirteen of them, round-half-away-from-zero was used; in the remaining six, round-half-to-even was used. I also discovered an anomaly in some environments: numbers like 0.15 — which look like halfway cases but are actually not when viewed in binary — may be rounded incorrectly. I’ll report my results in this article.

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Double Rounding Errors in Floating-Point Conversions

Double rounding is when a number is rounded twice, first from n0 digits to n1 digits, and then from n1 digits to n2 digits. Double rounding is often harmless, giving the same result as rounding once, directly from n0 digits to n2 digits. However, sometimes a doubly rounded result will be incorrect, in which case we say that a double rounding error has occurred.

For example, consider the 6-digit decimal number 7.23496. Rounded directly to 3 digits — using round-to-nearest, round half to even rounding — it’s 7.23; rounded first to 5 digits (7.2350) and then to 3 digits it’s 7.24. The value 7.24 is incorrect, reflecting a double rounding error.

In a computer, double rounding occurs in binary floating-point arithmetic; the typical example is a calculated result that’s rounded to fit into an x87 FPU extended precision register and then rounded again to fit into a double-precision variable. But I’ve discovered another context in which double rounding occurs: conversion from a decimal floating-point literal to a single-precision floating-point variable. The double rounding is from full-precision binary to double-precision, and then from double-precision to single-precision.

In this article, I’ll show example conversions in C that are tainted by double rounding errors, and how attaching the ‘f’ suffix to floating-point literals prevents them — in gcc C at least, but not in Visual C++!

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Displaying IEEE Doubles in Binary Scientific Notation

An IEEE double-precision floating-point number, or double, is a 64-bit encoding of a rational number. Internally, the 64 bits are broken into three fields: a 1-bit sign field, which represents positive or negative; an 11-bit exponent field, which represents a power of two; and a 52-bit fraction field, which represents the significant bits of the number. These three fields — together with an implicit leading 1 bit — represent a number in binary scientific notation, with 1 to 53 bits of precision.

For example, consider the decimal number 33.75. It converts to a double with a sign field of 0, an exponent field of 10000000100, and a fraction field of 0000111000000000000000000000000000000000000000000000. The 0 in the sign field means it’s a positive number (1 would mean it’s negative). The value of 10000000100 in the exponent field, which equals 1028 in decimal, means the exponent of the power of two is 5 (the exponent field value is offset, or biased, by 1023). The fraction field, when prefixed with an implicit leading 1, represents the binary fraction 1.0000111. Written in normalized binary scientific notation — following the convention that the fraction is written in binary and the power of two is written in decimal — 33.75 equals 1.0000111 x 25.

In this article, I’ll show you the C function I wrote to display a double in normalized binary scientific notation. This function is useful, for example, when verifying that decimal to floating-point conversions are correctly rounded.

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Quick and Dirty Decimal to Floating-Point Conversion

This little C program converts a decimal value — represented as a string — into a double-precision floating-point number:

#include <string.h>

int main (void)
{
 double intPart = 0, fracPart = 0, conversion;
 unsigned int i;
 char decimal[] = "3.14159";

 i = 0; /* Left to right */
 while (decimal[i] != '.') {
    intPart = intPart*10 + (decimal[i] - '0');
    i++;
   }

 i = strlen(decimal)-1; /* Right to left */
 while (decimal[i] != '.') {
    fracPart = (fracPart + (decimal[i] - '0'))/10;
    i--;
   }

 conversion = intPart + fracPart;
}

The conversion is done using the elegant Horner’s method, summing each digit according to its decimal place value. So why do I call it “quick and dirty?” Because the binary floating-point value it produces is not necessarily the closest approximation to the input decimal value — the so-called correctly rounded result. (Remember that most real numbers cannot be represented exactly in floating-point.) Most of the time it will produce the correctly rounded result, but sometimes it won’t — the result will be off in its least significant bit(s). There’s just not enough precision in floating-point to guarantee the result is correct every time.

I will demonstrate this program with different input values, some of which convert correctly, and some of which don’t. In the end, you’ll appreciate one reason why library functions like strtod() exist — to perform efficient, correctly rounded conversion.

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When Doubles Don’t Behave Like Doubles

In my article “When Floats Don’t Behave Like Floats” I explained how calculations involving single-precision floating-point variables may be done, under the covers, in double or extended precision. This leads to anomalies in expected results, which I demonstrated with two C programs — compiled with Microsoft Visual C++ and run on a 32-bit Intel Core Duo processor.

In this article, I’ll do a similar analysis for double-precision floating-point variables, showing how similar anomalies arise when extended precision calculations are done. I modified my two example programs to use doubles instead of floats. Interestingly, the doubles version of program 2 does not exhibit the anomaly. I’ll explain.

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When Floats Don’t Behave Like Floats

These two programs — compiled with Microsoft Visual C++ and run on a 32-bit Intel Core Duo processor — demonstrate an anomaly that occurs when using single-precision floating point variables:

Program 1

#include "stdio.h"
int main (void)
{
 float f1 = 0.1f, f2 = 3.0f, f3;

 f3 = f1 * f2;
 if (f3 != f1 * f2)
   printf("Not equal\n");
}

Prints “Not equal”.

Program 2

#include "stdio.h"
int main (void)
{
 float f1 = 0.7f, f2 = 10.0f, f3;
 int i1, i2;

 f3 = f1 * f2;
 i1 = (int)f3;
 i2 = (int)(f1 * f2);
 if (i1 != i2)
   printf("Not equal\n");
}

Prints “Not equal”.

In each case, f3 and f1 * f2 differ. But why? I’ll explain what’s going on.

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