Average Error: 38.0 → 12.9
Time: 19.6s
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 r28482 = 0.5;
        double r28483 = 2.0;
        double r28484 = re;
        double r28485 = r28484 * r28484;
        double r28486 = im;
        double r28487 = r28486 * r28486;
        double r28488 = r28485 + r28487;
        double r28489 = sqrt(r28488);
        double r28490 = r28489 - r28484;
        double r28491 = r28483 * r28490;
        double r28492 = sqrt(r28491);
        double r28493 = r28482 * r28492;
        return r28493;
}
↓
double f(double re, double im) {
        double r28494 = 0.5;
        double r28495 = re;
        double r28496 = im;
        double r28497 = hypot(r28495, r28496);
        double r28498 = r28497 - r28495;
        double r28499 = 2.0;
        double r28500 = r28498 * r28499;
        double r28501 = sqrt(r28500);
        double r28502 = r28494 * r28501;
        return r28502;
}

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 38.0

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

  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

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

Reproduce

herbie shell --seed 2019198 +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)))))