_multiplyComplex, real part

Average Error: 0.0 → 0.0

Time: 1.2s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r35458 = x_re;
        double r35459 = y_re;
        double r35460 = r35458 * r35459;
        double r35461 = x_im;
        double r35462 = y_im;
        double r35463 = r35461 * r35462;
        double r35464 = r35460 - r35463;
        return r35464;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r35465 = x_re;
        double r35466 = y_re;
        double r35467 = x_im;
        double r35468 = y_im;
        double r35469 = r35467 * r35468;
        double r35470 = -r35469;
        double r35471 = fma(r35465, r35466, r35470);
        return r35471;
}

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. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020100 +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)))