Average Error: 0.0 → 0.0
Time: 5.0s
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 r1568537 = x_re;
        double r1568538 = y_im;
        double r1568539 = r1568537 * r1568538;
        double r1568540 = x_im;
        double r1568541 = y_re;
        double r1568542 = r1568540 * r1568541;
        double r1568543 = r1568539 + r1568542;
        return r1568543;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1568544 = x_im;
        double r1568545 = y_re;
        double r1568546 = x_re;
        double r1568547 = y_im;
        double r1568548 = r1568546 * r1568547;
        double r1568549 = fma(r1568544, r1568545, r1568548);
        return r1568549;
}

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 2019138 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))