Average Error: 37.5 → 13.3
Time: 19.7s
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 r949961 = 0.5;
        double r949962 = 2.0;
        double r949963 = re;
        double r949964 = r949963 * r949963;
        double r949965 = im;
        double r949966 = r949965 * r949965;
        double r949967 = r949964 + r949966;
        double r949968 = sqrt(r949967);
        double r949969 = r949968 - r949963;
        double r949970 = r949962 * r949969;
        double r949971 = sqrt(r949970);
        double r949972 = r949961 * r949971;
        return r949972;
}
↓
double f(double re, double im) {
        double r949973 = re;
        double r949974 = im;
        double r949975 = hypot(r949973, r949974);
        double r949976 = r949975 - r949973;
        double r949977 = 2.0;
        double r949978 = r949976 * r949977;
        double r949979 = sqrt(r949978);
        double r949980 = 0.5;
        double r949981 = r949979 * r949980;
        return r949981;
}

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 37.5

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

  2. Simplified13.3

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

  3. Final simplification13.3

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

Reproduce

herbie shell --seed 2019163 +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)))))