Average Error: 0.0 → 0.0
Time: 2.7s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1500945 = x_re;
        double r1500946 = y_im;
        double r1500947 = r1500945 * r1500946;
        double r1500948 = x_im;
        double r1500949 = y_re;
        double r1500950 = r1500948 * r1500949;
        double r1500951 = r1500947 + r1500950;
        return r1500951;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1500952 = x_re;
        double r1500953 = y_im;
        double r1500954 = x_im;
        double r1500955 = y_re;
        double r1500956 = r1500954 * r1500955;
        double r1500957 = fma(r1500952, r1500953, r1500956);
        return r1500957;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))