Average Error: 37.7 → 13.6
Time: 25.3s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r551810 = 0.5;
        double r551811 = 2.0;
        double r551812 = re;
        double r551813 = r551812 * r551812;
        double r551814 = im;
        double r551815 = r551814 * r551814;
        double r551816 = r551813 + r551815;
        double r551817 = sqrt(r551816);
        double r551818 = r551817 - r551812;
        double r551819 = r551811 * r551818;
        double r551820 = sqrt(r551819);
        double r551821 = r551810 * r551820;
        return r551821;
}
↓
double f(double re, double im) {
        double r551822 = re;
        double r551823 = im;
        double r551824 = hypot(r551822, r551823);
        double r551825 = r551824 - r551822;
        double r551826 = 2.0;
        double r551827 = r551825 * r551826;
        double r551828 = sqrt(r551827);
        double r551829 = 0.5;
        double r551830 = r551828 * r551829;
        return r551830;
}

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. Simplified13.6

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0}}\]

  3. Final simplification13.6

    \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019128 +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)))))