_multiplyComplex, real part

Average Error: 0.0 → 0.0

Time: 1.1s

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 r58382 = x_re;
        double r58383 = y_re;
        double r58384 = r58382 * r58383;
        double r58385 = x_im;
        double r58386 = y_im;
        double r58387 = r58385 * r58386;
        double r58388 = r58384 - r58387;
        return r58388;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r58389 = x_re;
        double r58390 = y_re;
        double r58391 = x_im;
        double r58392 = y_im;
        double r58393 = r58391 * r58392;
        double r58394 = -r58393;
        double r58395 = fma(r58389, r58390, r58394);
        return r58395;
}

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 2019352 +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)))