Average Error: 38.4 → 12.9
Time: 36.4s
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 r786398 = 0.5;
        double r786399 = 2.0;
        double r786400 = re;
        double r786401 = r786400 * r786400;
        double r786402 = im;
        double r786403 = r786402 * r786402;
        double r786404 = r786401 + r786403;
        double r786405 = sqrt(r786404);
        double r786406 = r786405 - r786400;
        double r786407 = r786399 * r786406;
        double r786408 = sqrt(r786407);
        double r786409 = r786398 * r786408;
        return r786409;
}
↓
double f(double re, double im) {
        double r786410 = re;
        double r786411 = im;
        double r786412 = hypot(r786410, r786411);
        double r786413 = r786412 - r786410;
        double r786414 = 2.0;
        double r786415 = r786413 * r786414;
        double r786416 = sqrt(r786415);
        double r786417 = 0.5;
        double r786418 = r786416 * r786417;
        return r786418;
}

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. Simplified12.9

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

  3. Final simplification12.9

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

Reproduce

herbie shell --seed 2019168 +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)))))