Average Error: 38.6 → 13.1
Time: 10.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}
double f(double re, double im) {
        double r23292 = 0.5;
        double r23293 = 2.0;
        double r23294 = re;
        double r23295 = r23294 * r23294;
        double r23296 = im;
        double r23297 = r23296 * r23296;
        double r23298 = r23295 + r23297;
        double r23299 = sqrt(r23298);
        double r23300 = r23299 - r23294;
        double r23301 = r23293 * r23300;
        double r23302 = sqrt(r23301);
        double r23303 = r23292 * r23302;
        return r23303;
}
↓
double f(double re, double im) {
        double r23304 = 0.5;
        double r23305 = re;
        double r23306 = im;
        double r23307 = hypot(r23305, r23306);
        double r23308 = r23307 - r23305;
        double r23309 = 2.0;
        double r23310 = r23308 * r23309;
        double r23311 = sqrt(r23310);
        double r23312 = r23304 * r23311;
        return r23312;
}

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 38.6

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))