Average Error: 37.2 → 13.3
Time: 23.3s
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 r821299 = 0.5;
        double r821300 = 2.0;
        double r821301 = re;
        double r821302 = r821301 * r821301;
        double r821303 = im;
        double r821304 = r821303 * r821303;
        double r821305 = r821302 + r821304;
        double r821306 = sqrt(r821305);
        double r821307 = r821306 - r821301;
        double r821308 = r821300 * r821307;
        double r821309 = sqrt(r821308);
        double r821310 = r821299 * r821309;
        return r821310;
}
↓
double f(double re, double im) {
        double r821311 = re;
        double r821312 = im;
        double r821313 = hypot(r821311, r821312);
        double r821314 = r821313 - r821311;
        double r821315 = 2.0;
        double r821316 = r821314 * r821315;
        double r821317 = sqrt(r821316);
        double r821318 = 0.5;
        double r821319 = r821317 * r821318;
        return r821319;
}

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. 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 2019152 +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)))))