Average Error: 37.2 → 12.9
Time: 13.5s
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 r335724 = 0.5;
        double r335725 = 2.0;
        double r335726 = re;
        double r335727 = r335726 * r335726;
        double r335728 = im;
        double r335729 = r335728 * r335728;
        double r335730 = r335727 + r335729;
        double r335731 = sqrt(r335730);
        double r335732 = r335731 - r335726;
        double r335733 = r335725 * r335732;
        double r335734 = sqrt(r335733);
        double r335735 = r335724 * r335734;
        return r335735;
}
↓
double f(double re, double im) {
        double r335736 = re;
        double r335737 = im;
        double r335738 = hypot(r335736, r335737);
        double r335739 = r335738 - r335736;
        double r335740 = 2.0;
        double r335741 = r335739 * r335740;
        double r335742 = sqrt(r335741);
        double r335743 = 0.5;
        double r335744 = r335742 * r335743;
        return r335744;
}

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 37.2

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

  2. Simplified12.9

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

  3. Final simplification12.9

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

Reproduce

herbie shell --seed 2019156 +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)))))