_multiplyComplex, real part

Average Error: 0.5 → 0.2

Time: 2.8s

Precision: binary64

Math

\[x.re \cdot y.re - x.im \cdot y.im \]

↓

\[\mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right) \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- (* x.re y.re) (* x.im y.im)))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (fma x.re y.re (- (* x.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (x_46_re * y_46_re) - (x_46_im * y_46_im);
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return fma(x_46_re, y_46_re, -(x_46_im * y_46_im));
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

1. Initial program 0.5

   \[x.re \cdot y.re - x.im \cdot y.im \]

2. Applied fma-neg_binary64 0.2

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

3. Final simplification 0.2

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

Reproduce

herbie shell --seed 2022067 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))