Average Error: 37.2 → 12.9
Time: 17.1s
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 r652968 = 0.5;
        double r652969 = 2.0;
        double r652970 = re;
        double r652971 = r652970 * r652970;
        double r652972 = im;
        double r652973 = r652972 * r652972;
        double r652974 = r652971 + r652973;
        double r652975 = sqrt(r652974);
        double r652976 = r652975 - r652970;
        double r652977 = r652969 * r652976;
        double r652978 = sqrt(r652977);
        double r652979 = r652968 * r652978;
        return r652979;
}
↓
double f(double re, double im) {
        double r652980 = re;
        double r652981 = im;
        double r652982 = hypot(r652980, r652981);
        double r652983 = r652982 - r652980;
        double r652984 = 2.0;
        double r652985 = r652983 * r652984;
        double r652986 = sqrt(r652985);
        double r652987 = 0.5;
        double r652988 = r652986 * r652987;
        return r652988;
}

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 2019165 +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)))))