#Fused Multiply-Add S2

#Fused Multiply-Add

Stage: 2

Champion: Waldemar Horwat @waldemarhorwat

Draft spec

Fused multiply-add is a three-operand IEEE floating-point operation that performs the computation $x × y + z$ with no intermediate rounding of the product — only the final result is rounded. It is a standard operation supported by modern floating-point hardware and mandated by IEEE 754-2019.

#Background

The IEEE 754 standard mandates, in addition to other kinds of operations, six basic arithmetic operations: +, -, ×, /, square root, and fused multiply-add. The first five were original, while fused multiply-add was added in IEEE 754-2008.

ECMAScript has implemented the first five of these since the beginning, but has not included fused multiply-add when the IEEE 754 spec was updated in 2008. We propose adding it to the language.

Fused multiply-add is supported with IEEE 754 semantics in other major programming languages, including:

• C/C++: fma, fmaf, fmal
• Java: Math.Fma
• C#: Math.FusedMultiplyAdd
• Python: math.fma
• Rust: mul_add
• Swift: addProduct, addingProduct

#Usage

Fused multiply-add, a basic floating-point instruction supported by modern hardware, is essential for correctness and performance in many numerical algorithms. Its lack in ECMAScript presents a challenge to anyone trying to adapt code that makes use of it in other languages — fused multiply-add is difficult to implement as a library, requiring hundreds of lines of slow and tricky code.

Common uses include computing dot products, matrix multiplication, evaluating polynomials, and neural network applications.

For a simple example of how one can use fused multiply-add, here's how we can compute the exact product of Numbers $a$ and $b$ with no rounding:

let high = a * b (using standard IEEE 754 multiplication)

let err = Math.fma(a, b, -high)

At this point, barring overflow or underflow of Number's exponent range, $MV(a) × MV(b) = MV(high) + MV(err)$, with $high$ containing the most significant bits of the result and $err$ containing the least significant bits of the result.