Decimal Floating-Point (dfloat): Base-10 Arithmetic Without Binary Rounding

Why

The value 0.1 cannot be represented exactly in binary floating-point. Neither can 0.2, 0.3, or any decimal fraction whose denominator is not a power of two. This means that 0.1 + 0.2 != 0.3 in IEEE-754 — a fact that has caused countless bugs in financial software, tax computation, and any domain where humans expect base-10 arithmetic to behave like base-10 arithmetic.

The dfloat type represents numbers in base-10, where decimal fractions are exact. The value 0.1 is stored as the digit 1 with a decimal exponent of -1, with no binary rounding error. This makes dfloat the correct choice for any computation that must match pencil-and-paper decimal arithmetic.

What

dfloat<ndigits, es, bt> is a decimal floating-point number:

Parameter	Type	Default	Description
ndigits	unsigned	—	Number of significant decimal digits
es	unsigned	—	Exponent field width (binary-encoded decimal exponent)
bt	typename	uint8_t	Storage block type

Key Properties

• Base-10 representation: decimal fractions are exact
• Decimal digits: ndigits significant digits of precision
• Decimal exponent: value = significand x 10^exponent
• IEEE 754-2008 compatible: targets decimal interchange format (DPD/BID)
• No binary rounding error for values expressible as d x 10^e

Status

The dfloat type is currently a skeleton implementation with the core infrastructure defined. The encoding format and arithmetic pipeline are specified, but full arithmetic operations are still under development.

How It Works

Unlike binary floating-point (which stores significand x 2^exponent), dfloat stores significand x 10^exponent. Each decimal digit is encoded in the significand field, and the exponent represents a power of 10. This means:

• 0.1 = 1 x 10^-1 (exact)
• 0.01 = 1 x 10^-2 (exact)
• 9.99 = 999 x 10^-2 (exact)
• 1/3 = still inexact (0.333… must be truncated)

The encoding follows the IEEE 754-2008 decimal format principles, potentially using Densely Packed Decimal (DPD) encoding for efficient digit storage.

How to Use It

Include

#include <universal/number/dfloat/dfloat.hpp>
using namespace sw::universal;

Basic Usage

dfloat<7, 8> price(19.99);    // 7 significant decimal digits
dfloat<7, 8> tax_rate(0.075);  // 7.5% tax, exact in decimal
dfloat<7, 8> tax = price * tax_rate;
dfloat<7, 8> total = price + tax;
// No binary rounding: 19.99 and 0.075 are represented exactly

Problems It Solves

Problem	How dfloat Solves It
0.1 + 0.2 != 0.3 in binary floating-point	Decimal fractions are exact in base-10
Financial calculations accumulate penny rounding errors	Exact representation of dollars and cents
Tax computation must match regulatory decimal rules	Base-10 arithmetic matches pencil-and-paper rules
Database NUMERIC/DECIMAL types lose precision in C++	dfloat preserves decimal precision natively
Currency conversion chains compound binary errors	Decimal intermediate values stay exact
Regulatory compliance requires decimal rounding rules	Native decimal rounding (not binary approximation)