Average Error: 37.6 → 13.4
Time: 18.9s
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 r537062 = 0.5;
        double r537063 = 2.0;
        double r537064 = re;
        double r537065 = r537064 * r537064;
        double r537066 = im;
        double r537067 = r537066 * r537066;
        double r537068 = r537065 + r537067;
        double r537069 = sqrt(r537068);
        double r537070 = r537069 - r537064;
        double r537071 = r537063 * r537070;
        double r537072 = sqrt(r537071);
        double r537073 = r537062 * r537072;
        return r537073;
}
↓
double f(double re, double im) {
        double r537074 = re;
        double r537075 = im;
        double r537076 = hypot(r537074, r537075);
        double r537077 = r537076 - r537074;
        double r537078 = 2.0;
        double r537079 = r537077 * r537078;
        double r537080 = sqrt(r537079);
        double r537081 = 0.5;
        double r537082 = r537080 * r537081;
        return r537082;
}

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 37.6

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

  2. Simplified13.4

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

  3. Final simplification13.4

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

Reproduce

herbie shell --seed 2019130 +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)))))