math.abs on complex (squared)

Average Error: 0.0 → 0.0

Time: 578.0ms

Precision: binary64

Math

\[re \cdot re + im \cdot im \]

↓

\[\mathsf{fma}\left(re, re, im \cdot im\right) \]

FPCore

(FPCore modulus_sqr (re im) :precision binary64 (+ (* re re) (* im im)))

↓

(FPCore modulus_sqr (re im) :precision binary64 (fma re re (* im im)))

C

double modulus_sqr(double re, double im) {
	return (re * re) + (im * im);
}

↓

double modulus_sqr(double re, double im) {
	return fma(re, re, (im * im));
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.0

   \[re \cdot re + im \cdot im \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, im \cdot im\right)} \]

3. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(re, re, im \cdot im\right) \]

Reproduce

herbie shell --seed 2022067 
(FPCore modulus_sqr (re im)
  :name "math.abs on complex (squared)"
  :precision binary64
  (+ (* re re) (* im im)))