Average Error: 37.7 → 13.3
Time: 11.3s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r65813 = 0.5;
        double r65814 = 2.0;
        double r65815 = re;
        double r65816 = r65815 * r65815;
        double r65817 = im;
        double r65818 = r65817 * r65817;
        double r65819 = r65816 + r65818;
        double r65820 = sqrt(r65819);
        double r65821 = r65820 - r65815;
        double r65822 = r65814 * r65821;
        double r65823 = sqrt(r65822);
        double r65824 = r65813 * r65823;
        return r65824;
}
↓
double f(double re, double im) {
        double r65825 = re;
        double r65826 = im;
        double r65827 = hypot(r65825, r65826);
        double r65828 = r65827 - r65825;
        double r65829 = 2.0;
        double r65830 = r65828 * r65829;
        double r65831 = sqrt(r65830);
        double r65832 = 0.5;
        double r65833 = r65831 * r65832;
        return r65833;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 37.7

    \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

  2. Simplified13.3

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0}}\]

  3. Final simplification13.3

    \[\leadsto \sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))