_multiplyComplex, real part

Average Error: 0.0 → 0.0

Time: 2.2s

Precision: 64

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

Math

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

↓

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

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r94310 = x_re;
        double r94311 = y_re;
        double r94312 = r94310 * r94311;
        double r94313 = x_im;
        double r94314 = y_im;
        double r94315 = r94313 * r94314;
        double r94316 = r94312 - r94315;
        return r94316;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r94317 = x_re;
        double r94318 = y_re;
        double r94319 = x_im;
        double r94320 = y_im;
        double r94321 = r94319 * r94320;
        double r94322 = -r94321;
        double r94323 = fma(r94317, r94318, r94322);
        return r94323;
}

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 2020042 +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)))