Average Error: 39.0 → 13.7
Time: 19.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}
double f(double re, double im) {
        double r24799 = 0.5;
        double r24800 = 2.0;
        double r24801 = re;
        double r24802 = r24801 * r24801;
        double r24803 = im;
        double r24804 = r24803 * r24803;
        double r24805 = r24802 + r24804;
        double r24806 = sqrt(r24805);
        double r24807 = r24806 - r24801;
        double r24808 = r24800 * r24807;
        double r24809 = sqrt(r24808);
        double r24810 = r24799 * r24809;
        return r24810;
}
↓
double f(double re, double im) {
        double r24811 = 0.5;
        double r24812 = re;
        double r24813 = im;
        double r24814 = hypot(r24812, r24813);
        double r24815 = r24814 - r24812;
        double r24816 = 2.0;
        double r24817 = r24815 * r24816;
        double r24818 = sqrt(r24817);
        double r24819 = r24811 * r24818;
        return r24819;
}

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 39.0

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

  2. Simplified13.7

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

  3. Final simplification13.7

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))