Average Error: 39.0 → 13.3
Time: 8.7s
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 r24860 = 0.5;
        double r24861 = 2.0;
        double r24862 = re;
        double r24863 = r24862 * r24862;
        double r24864 = im;
        double r24865 = r24864 * r24864;
        double r24866 = r24863 + r24865;
        double r24867 = sqrt(r24866);
        double r24868 = r24867 - r24862;
        double r24869 = r24861 * r24868;
        double r24870 = sqrt(r24869);
        double r24871 = r24860 * r24870;
        return r24871;
}
↓
double f(double re, double im) {
        double r24872 = 0.5;
        double r24873 = re;
        double r24874 = im;
        double r24875 = hypot(r24873, r24874);
        double r24876 = r24875 - r24873;
        double r24877 = 2.0;
        double r24878 = r24876 * r24877;
        double r24879 = sqrt(r24878);
        double r24880 = r24872 * r24879;
        return r24880;
}

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. Simplified13.3

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

  3. Final simplification13.3

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

Reproduce

herbie shell --seed 2020047 +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)))))