Average Error: 0.0 → 0.0
Time: 2.3s
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1135373 = x_re;
        double r1135374 = y_re;
        double r1135375 = r1135373 * r1135374;
        double r1135376 = x_im;
        double r1135377 = y_im;
        double r1135378 = r1135376 * r1135377;
        double r1135379 = r1135375 - r1135378;
        return r1135379;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1135380 = x_re;
        double r1135381 = y_re;
        double r1135382 = x_im;
        double r1135383 = y_im;
        double r1135384 = r1135382 * r1135383;
        double r1135385 = -r1135384;
        double r1135386 = fma(r1135380, r1135381, r1135385);
        return r1135386;
}

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. Using strategy rm

  3. Applied fma-neg0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))