Average Error: 38.2 → 13.5
Time: 15.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5
double f(double re, double im) {
        double r29252 = 0.5;
        double r29253 = 2.0;
        double r29254 = re;
        double r29255 = r29254 * r29254;
        double r29256 = im;
        double r29257 = r29256 * r29256;
        double r29258 = r29255 + r29257;
        double r29259 = sqrt(r29258);
        double r29260 = r29259 - r29254;
        double r29261 = r29253 * r29260;
        double r29262 = sqrt(r29261);
        double r29263 = r29252 * r29262;
        return r29263;
}
↓
double f(double re, double im) {
        double r29264 = re;
        double r29265 = im;
        double r29266 = hypot(r29264, r29265);
        double r29267 = r29266 - r29264;
        double r29268 = 2.0;
        double r29269 = r29267 * r29268;
        double r29270 = sqrt(r29269);
        double r29271 = 0.5;
        double r29272 = r29270 * r29271;
        return r29272;
}

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 38.2

    \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

  2. Simplified13.5

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}}\]

  3. Final simplification13.5

    \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\]

Reproduce

herbie shell --seed 2019194 +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)))))