Average Error: 0.0 → 0.0
Time: 2.8s
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 r44143 = x_re;
        double r44144 = y_re;
        double r44145 = r44143 * r44144;
        double r44146 = x_im;
        double r44147 = y_im;
        double r44148 = r44146 * r44147;
        double r44149 = r44145 - r44148;
        return r44149;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r44150 = x_re;
        double r44151 = y_re;
        double r44152 = x_im;
        double r44153 = y_im;
        double r44154 = r44152 * r44153;
        double r44155 = -r44154;
        double r44156 = fma(r44150, r44151, r44155);
        return r44156;
}

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 2020060 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))