Average Error: 38.6 → 13.2
Time: 21.6s
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 r30973 = 0.5;
        double r30974 = 2.0;
        double r30975 = re;
        double r30976 = r30975 * r30975;
        double r30977 = im;
        double r30978 = r30977 * r30977;
        double r30979 = r30976 + r30978;
        double r30980 = sqrt(r30979);
        double r30981 = r30980 - r30975;
        double r30982 = r30974 * r30981;
        double r30983 = sqrt(r30982);
        double r30984 = r30973 * r30983;
        return r30984;
}
↓
double f(double re, double im) {
        double r30985 = 0.5;
        double r30986 = re;
        double r30987 = im;
        double r30988 = hypot(r30986, r30987);
        double r30989 = r30988 - r30986;
        double r30990 = 2.0;
        double r30991 = r30989 * r30990;
        double r30992 = sqrt(r30991);
        double r30993 = r30985 * r30992;
        return r30993;
}

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 38.6

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

  2. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2019323 +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)))))