_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 r39667 = x_re;
        double r39668 = y_re;
        double r39669 = r39667 * r39668;
        double r39670 = x_im;
        double r39671 = y_im;
        double r39672 = r39670 * r39671;
        double r39673 = r39669 - r39672;
        return r39673;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r39674 = x_re;
        double r39675 = y_re;
        double r39676 = x_im;
        double r39677 = y_im;
        double r39678 = r39676 * r39677;
        double r39679 = -r39678;
        double r39680 = fma(r39674, r39675, r39679);
        return r39680;
}

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 2020036 +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)))