Average Error: 37.6 → 13.4
Time: 19.4s
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 r612183 = 0.5;
        double r612184 = 2.0;
        double r612185 = re;
        double r612186 = r612185 * r612185;
        double r612187 = im;
        double r612188 = r612187 * r612187;
        double r612189 = r612186 + r612188;
        double r612190 = sqrt(r612189);
        double r612191 = r612190 - r612185;
        double r612192 = r612184 * r612191;
        double r612193 = sqrt(r612192);
        double r612194 = r612183 * r612193;
        return r612194;
}
↓
double f(double re, double im) {
        double r612195 = re;
        double r612196 = im;
        double r612197 = hypot(r612195, r612196);
        double r612198 = r612197 - r612195;
        double r612199 = 2.0;
        double r612200 = r612198 * r612199;
        double r612201 = sqrt(r612200);
        double r612202 = 0.5;
        double r612203 = r612201 * r612202;
        return r612203;
}

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 2019132 +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)))))