Average Error: 0.0 → 0.0
Time: 10.4s
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 r2102375 = x_re;
        double r2102376 = y_re;
        double r2102377 = r2102375 * r2102376;
        double r2102378 = x_im;
        double r2102379 = y_im;
        double r2102380 = r2102378 * r2102379;
        double r2102381 = r2102377 - r2102380;
        return r2102381;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2102382 = x_re;
        double r2102383 = y_re;
        double r2102384 = x_im;
        double r2102385 = y_im;
        double r2102386 = r2102384 * r2102385;
        double r2102387 = -r2102386;
        double r2102388 = fma(r2102382, r2102383, r2102387);
        return r2102388;
}

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 2019162 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))