## strtod()’s Initial Decimal to Floating-Point Approximation

David Gay’s strtod() function does decimal to floating-point conversion using both IEEE double-precision floating-point arithmetic and arbitrary-precision integer arithmetic. For some inputs, a simple IEEE floating-point calculation suffices to produce the correct result; for other inputs, a combination of IEEE arithmetic and arbitrary-precision arithmetic is required. In the latter case, IEEE arithmetic is used to calculate an approximation to the correct result, which is then refined using arbitrary-precision arithmetic. In this article, I’ll describe the approximation calculation, which is based on a form of binary exponentiation.

## Nondeterministic Floating-Point Conversions in Java

Recently I discovered that Java converts some very small decimal numbers to double-precision floating-point incorrectly. While investigating that bug, I stumbled upon something very strange: Java’s decimal to floating-point conversion routine, Double.parseDouble(), sometimes returns two different results for the same decimal string. The culprit appears to be just-in-time compilation of Double.parseDouble() into SSE instructions, which exposes an architecture-dependent bug in Java’s conversion algorithm — and another real-world example of a double rounding on underflow error. I’ll describe the problem, and take you through the detective work to find its cause.

## A Closer Look at the Java 2.2250738585072012e-308 Bug

Java’s decimal to floating-point conversion routine, the doubleValue() method of its FloatingDecimal class, goes into an infinite loop when converting the decimal string 2.2250738585072012e-308 to double-precision binary floating-point. I took a closer look at the bug, by tracing the doubleValue() method in the Eclipse IDE for Java (thanks to Konstantin Preißer for helping me set that up). What I found was that our initial analysis of the bug was wrong; what actually happens is that doubleValue()’s correction loop oscillates between two values, 0x1p-1022 and 0x0.fffffffffffffp-1022.

## 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.

## 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.

## 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.

## 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.

## 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!

## 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.

## 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.

## 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++!

## 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.