_multiplyComplex, imaginary part

Average Error: 0.0 → 0.0

Time: 11.5s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r10003 = x_re;
        double r10004 = y_im;
        double r10005 = r10003 * r10004;
        double r10006 = x_im;
        double r10007 = y_re;
        double r10008 = r10006 * r10007;
        double r10009 = r10005 + r10008;
        return r10009;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r10010 = y_re;
        double r10011 = x_im;
        double r10012 = y_im;
        double r10013 = x_re;
        double r10014 = r10012 * r10013;
        double r10015 = fma(r10010, r10011, r10014);
        return r10015;
}

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

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)}\]
3. Using strategy rm

4. Applied fma-udef 0.0

   \[\leadsto \color{blue}{x.re \cdot y.im + x.im \cdot y.re}\]

5. Simplified 0.0

   \[\leadsto \color{blue}{y.im \cdot x.re} + x.im \cdot y.re\]

6. Taylor expanded around inf 0.0

   \[\leadsto \color{blue}{y.re \cdot x.im + y.im \cdot x.re}\]

7. Simplified 0.0

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

8. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019310 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))