Average Error: 0.0 → 0.0
Time: 3.7s
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 r1899339 = x_re;
        double r1899340 = y_re;
        double r1899341 = r1899339 * r1899340;
        double r1899342 = x_im;
        double r1899343 = y_im;
        double r1899344 = r1899342 * r1899343;
        double r1899345 = r1899341 - r1899344;
        return r1899345;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1899346 = x_re;
        double r1899347 = y_re;
        double r1899348 = x_im;
        double r1899349 = y_im;
        double r1899350 = r1899348 * r1899349;
        double r1899351 = -r1899350;
        double r1899352 = fma(r1899346, r1899347, r1899351);
        return r1899352;
}

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 2019158 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))