Average Error: 0.0 → 0.0
Time: 5.6s
Precision: binary64
\[x.re \cdot y.re - x.im \cdot y.im \]
↓
\[\mathsf{fma}\left(x.re, y.re, -y.im \cdot x.im\right) + \mathsf{fma}\left(-y.im, x.im, y.im \cdot x.im\right) \]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, -y.im \cdot x.im\right) + \mathsf{fma}\left(-y.im, x.im, y.im \cdot x.im\right)

(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 (- (* y.im x.im))) (fma (- y.im) x.im (* y.im x.im))))
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, -(y_46_im * x_46_im)) + fma(-y_46_im, x_46_im, (y_46_im * x_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 0.0

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

  2. Applied prod-diff_binary640.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2021275 
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))