Average Error: 0.0 → 0.0
Time: 5.8s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.im, y.re, x.re \cdot y.im\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.im, y.re, x.re \cdot y.im\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1991198 = x_re;
        double r1991199 = y_im;
        double r1991200 = r1991198 * r1991199;
        double r1991201 = x_im;
        double r1991202 = y_re;
        double r1991203 = r1991201 * r1991202;
        double r1991204 = r1991200 + r1991203;
        return r1991204;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1991205 = x_im;
        double r1991206 = y_re;
        double r1991207 = x_re;
        double r1991208 = y_im;
        double r1991209 = r1991207 * r1991208;
        double r1991210 = fma(r1991205, r1991206, r1991209);
        return r1991210;
}

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

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

  2. Simplified0.0

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

  3. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{y.re \cdot x.im + y.im \cdot x.re}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))