Average Error: 38.5 → 13.1
Time: 15.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5
double f(double re, double im) {
        double r31726 = 0.5;
        double r31727 = 2.0;
        double r31728 = re;
        double r31729 = r31728 * r31728;
        double r31730 = im;
        double r31731 = r31730 * r31730;
        double r31732 = r31729 + r31731;
        double r31733 = sqrt(r31732);
        double r31734 = r31733 - r31728;
        double r31735 = r31727 * r31734;
        double r31736 = sqrt(r31735);
        double r31737 = r31726 * r31736;
        return r31737;
}
↓
double f(double re, double im) {
        double r31738 = re;
        double r31739 = im;
        double r31740 = hypot(r31738, r31739);
        double r31741 = r31740 - r31738;
        double r31742 = 2.0;
        double r31743 = r31741 * r31742;
        double r31744 = sqrt(r31743);
        double r31745 = 0.5;
        double r31746 = r31744 * r31745;
        return r31746;
}

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 38.5

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

  2. Simplified13.1

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}}\]

  3. Final simplification13.1

    \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\]

Reproduce

herbie shell --seed 2019196 +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)))))