Average Error: 38.1 → 13.3
Time: 18.8s
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 r499817 = 0.5;
        double r499818 = 2.0;
        double r499819 = re;
        double r499820 = r499819 * r499819;
        double r499821 = im;
        double r499822 = r499821 * r499821;
        double r499823 = r499820 + r499822;
        double r499824 = sqrt(r499823);
        double r499825 = r499824 - r499819;
        double r499826 = r499818 * r499825;
        double r499827 = sqrt(r499826);
        double r499828 = r499817 * r499827;
        return r499828;
}
↓
double f(double re, double im) {
        double r499829 = re;
        double r499830 = im;
        double r499831 = hypot(r499829, r499830);
        double r499832 = r499831 - r499829;
        double r499833 = 2.0;
        double r499834 = r499832 * r499833;
        double r499835 = sqrt(r499834);
        double r499836 = 0.5;
        double r499837 = r499835 * r499836;
        return r499837;
}

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 38.1

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

  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 2019143 +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)))))