Average Error: 37.5 → 13.3
Time: 18.1s
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 r844376 = 0.5;
        double r844377 = 2.0;
        double r844378 = re;
        double r844379 = r844378 * r844378;
        double r844380 = im;
        double r844381 = r844380 * r844380;
        double r844382 = r844379 + r844381;
        double r844383 = sqrt(r844382);
        double r844384 = r844383 - r844378;
        double r844385 = r844377 * r844384;
        double r844386 = sqrt(r844385);
        double r844387 = r844376 * r844386;
        return r844387;
}
↓
double f(double re, double im) {
        double r844388 = re;
        double r844389 = im;
        double r844390 = hypot(r844388, r844389);
        double r844391 = r844390 - r844388;
        double r844392 = 2.0;
        double r844393 = r844391 * r844392;
        double r844394 = sqrt(r844393);
        double r844395 = 0.5;
        double r844396 = r844394 * r844395;
        return r844396;
}

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 37.5

    \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

  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 2019163 +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)))))