Average Error: 39.0 → 13.3
Time: 10.1s
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 r29596 = 0.5;
        double r29597 = 2.0;
        double r29598 = re;
        double r29599 = r29598 * r29598;
        double r29600 = im;
        double r29601 = r29600 * r29600;
        double r29602 = r29599 + r29601;
        double r29603 = sqrt(r29602);
        double r29604 = r29603 - r29598;
        double r29605 = r29597 * r29604;
        double r29606 = sqrt(r29605);
        double r29607 = r29596 * r29606;
        return r29607;
}
↓
double f(double re, double im) {
        double r29608 = 0.5;
        double r29609 = re;
        double r29610 = im;
        double r29611 = hypot(r29609, r29610);
        double r29612 = r29611 - r29609;
        double r29613 = 2.0;
        double r29614 = r29612 * r29613;
        double r29615 = sqrt(r29614);
        double r29616 = r29608 * r29615;
        return r29616;
}

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 39.0

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

  2. Simplified13.3

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

  3. Final simplification13.3

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

Reproduce

herbie shell --seed 2020047 +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)))))