Average Error: 1.1 → 1.1
Time: 4.7s
Precision: 64
\[\frac{\left(\frac{\left(x.re \cdot y.re\right)}{\left(x.im \cdot y.im\right)}\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
↓
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
\frac{\left(\frac{\left(x.re \cdot y.re\right)}{\left(x.im \cdot y.im\right)}\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}
↓
\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1826314 = x_re;
        double r1826315 = y_re;
        double r1826316 = r1826314 * r1826315;
        double r1826317 = x_im;
        double r1826318 = y_im;
        double r1826319 = r1826317 * r1826318;
        double r1826320 = r1826316 + r1826319;
        double r1826321 = r1826315 * r1826315;
        double r1826322 = r1826318 * r1826318;
        double r1826323 = r1826321 + r1826322;
        double r1826324 = r1826320 / r1826323;
        return r1826324;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1826325 = x_re;
        double r1826326 = y_re;
        double r1826327 = r1826325 * r1826326;
        double r1826328 = x_im;
        double r1826329 = y_im;
        double r1826330 = r1826328 * r1826329;
        double r1826331 = r1826327 + r1826330;
        double r1826332 = r1826326 * r1826326;
        double r1826333 = r1826329 * r1826329;
        double r1826334 = r1826332 + r1826333;
        double r1826335 = r1826331 / r1826334;
        return r1826335;
}

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 1.1

    \[\frac{\left(\frac{\left(x.re \cdot y.re\right)}{\left(x.im \cdot y.im\right)}\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]

  2. Final simplification1.1

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  (/.p16 (+.p16 (*.p16 x.re y.re) (*.p16 x.im y.im)) (+.p16 (*.p16 y.re y.re) (*.p16 y.im y.im))))