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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Incorrect Directed Conversions in David Gay’s strtod()

For correctly rounded decimal to floating-point conversions, many open source projects rely on David Gay’s strtod() function. In the default rounding mode, IEEE 754 round-to-nearest, this function is known to give correct results (notwithstanding recent bugs, which have been fixed). However, in the less frequently used IEEE 754 directed rounding modes — round toward positive infinity, round toward negative infinity, and round toward zero — strtod() gives incorrectly rounded results for some inputs.

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Visual C++ and GLIBC strtod() Ignore Rounding Mode

When a decimal number is converted to a binary floating-point number, the floating-point number, in general, is only an approximation to the decimal number. Large integers, and most decimal fractions, require more significant bits than can be represented in the floating-point format. This means the decimal number must be rounded, to one of the two floating-point numbers that surround it.

Common practice considers a decimal number correctly rounded when the nearest of the two floating-point numbers is chosen (and when both are equally near, when the one with significant bit number 53 equal to 0 is chosen). This makes sense intuitively, and also reflects the default IEEE 754 rounding mode — round-to-nearest. However, there are three other IEEE 754 rounding modes, which allow for directed rounding: round toward positive infinity, round toward negative infinity, and round toward zero. For a conversion to be considered truly correctly rounded, it must honor all four rounding modes — whichever is currently in effect.

I evaluated the Visual C++ and glibc strtod() functions under the three directed rounding modes, like I did for round-to-nearest mode in my articles “Incorrectly Rounded Conversions in Visual C++” and “Incorrectly Rounded Conversions in GCC and GLIBC”. What I discovered was this: they only convert correctly about half the time — pure chance! — because they ignore the rounding mode altogether.

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Incorrectly Rounded Conversions in GCC and GLIBC

Visual C++ rounds some decimal to double-precision floating-point conversions incorrectly, but it’s not alone; the gcc C compiler and the glibc strtod() function do the same. In this article, I’ll show examples of incorrect conversions in gcc and glibc, and I’ll present a C program that demonstrates the errors.

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Incorrectly Rounded Conversions in Visual C++

In my analysis of decimal to floating-point conversion I noted an example that was converted incorrectly by the Microsoft Visual C++ compiler. I’ve found more examples — including a class of examples — that it converts incorrectly. I will analyze those examples in this article.

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