Average Error: 0.0 → 0.0
Time: 801.0ms
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r95480 = x_re;
        double r95481 = y_re;
        double r95482 = r95480 * r95481;
        double r95483 = x_im;
        double r95484 = y_im;
        double r95485 = r95483 * r95484;
        double r95486 = r95482 - r95485;
        return r95486;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r95487 = x_re;
        double r95488 = y_re;
        double r95489 = x_im;
        double r95490 = y_im;
        double r95491 = r95489 * r95490;
        double r95492 = -r95491;
        double r95493 = fma(r95487, r95488, r95492);
        return r95493;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))