Average Error: 1.1 → 1.1
Time: 4.9s
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 r676943 = x_re;
        double r676944 = y_re;
        double r676945 = r676943 * r676944;
        double r676946 = x_im;
        double r676947 = y_im;
        double r676948 = r676946 * r676947;
        double r676949 = r676945 + r676948;
        double r676950 = r676944 * r676944;
        double r676951 = r676947 * r676947;
        double r676952 = r676950 + r676951;
        double r676953 = r676949 / r676952;
        return r676953;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r676954 = x_re;
        double r676955 = y_re;
        double r676956 = r676954 * r676955;
        double r676957 = x_im;
        double r676958 = y_im;
        double r676959 = r676957 * r676958;
        double r676960 = r676956 + r676959;
        double r676961 = r676955 * r676955;
        double r676962 = r676958 * r676958;
        double r676963 = r676961 + r676962;
        double r676964 = r676960 / r676963;
        return r676964;
}

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 2019124 
(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))))