Average Error: 37.5 → 13.5
Time: 24.6s
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 r632685 = 0.5;
        double r632686 = 2.0;
        double r632687 = re;
        double r632688 = r632687 * r632687;
        double r632689 = im;
        double r632690 = r632689 * r632689;
        double r632691 = r632688 + r632690;
        double r632692 = sqrt(r632691);
        double r632693 = r632692 - r632687;
        double r632694 = r632686 * r632693;
        double r632695 = sqrt(r632694);
        double r632696 = r632685 * r632695;
        return r632696;
}
↓
double f(double re, double im) {
        double r632697 = re;
        double r632698 = im;
        double r632699 = hypot(r632697, r632698);
        double r632700 = r632699 - r632697;
        double r632701 = 2.0;
        double r632702 = r632700 * r632701;
        double r632703 = sqrt(r632702);
        double r632704 = 0.5;
        double r632705 = r632703 * r632704;
        return r632705;
}

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. Simplified13.5

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0}}\]

  3. Final simplification13.5

    \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019139 +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)))))