_multiplyComplex, real part

Average Error: 0.0 → 0.0

Time: 9.3s

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 r60489 = x_re;
        double r60490 = y_re;
        double r60491 = r60489 * r60490;
        double r60492 = x_im;
        double r60493 = y_im;
        double r60494 = r60492 * r60493;
        double r60495 = r60491 - r60494;
        return r60495;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r60496 = x_re;
        double r60497 = y_re;
        double r60498 = y_im;
        double r60499 = x_im;
        double r60500 = r60498 * r60499;
        double r60501 = -r60500;
        double r60502 = fma(r60496, r60497, r60501);
        return r60502;
}

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 2019351 +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)))