Average Error: 38.4 → 13.2
Time: 19.4s
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 r26280 = 0.5;
        double r26281 = 2.0;
        double r26282 = re;
        double r26283 = r26282 * r26282;
        double r26284 = im;
        double r26285 = r26284 * r26284;
        double r26286 = r26283 + r26285;
        double r26287 = sqrt(r26286);
        double r26288 = r26287 - r26282;
        double r26289 = r26281 * r26288;
        double r26290 = sqrt(r26289);
        double r26291 = r26280 * r26290;
        return r26291;
}
↓
double f(double re, double im) {
        double r26292 = 0.5;
        double r26293 = re;
        double r26294 = im;
        double r26295 = hypot(r26293, r26294);
        double r26296 = r26295 - r26293;
        double r26297 = 2.0;
        double r26298 = r26296 * r26297;
        double r26299 = sqrt(r26298);
        double r26300 = r26292 * r26299;
        return r26300;
}

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 38.4

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

  2. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2019326 +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)))))