Average Error: 0.0 → 0.0
Time: 3.4s
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r933178 = x_re;
        double r933179 = y_re;
        double r933180 = r933178 * r933179;
        double r933181 = x_im;
        double r933182 = y_im;
        double r933183 = r933181 * r933182;
        double r933184 = r933180 - r933183;
        return r933184;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r933185 = x_re;
        double r933186 = y_re;
        double r933187 = x_im;
        double r933188 = y_im;
        double r933189 = r933187 * r933188;
        double r933190 = -r933189;
        double r933191 = fma(r933185, r933186, r933190);
        return r933191;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))