_multiplyComplex, real part

Average Error: 0.0 → 0.0

Time: 2.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r88265 = x_re;
        double r88266 = y_re;
        double r88267 = r88265 * r88266;
        double r88268 = x_im;
        double r88269 = y_im;
        double r88270 = r88268 * r88269;
        double r88271 = r88267 - r88270;
        return r88271;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r88272 = x_re;
        double r88273 = y_re;
        double r88274 = y_im;
        double r88275 = x_im;
        double r88276 = r88274 * r88275;
        double r88277 = -r88276;
        double r88278 = fma(r88272, r88273, r88277);
        return r88278;
}

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-neg 0.0

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

4. Simplified 0.0

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

5. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019350 +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)))