Average Error: 38.4 → 13.2
Time: 19.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5
double f(double re, double im) {
        double r591812 = 0.5;
        double r591813 = 2.0;
        double r591814 = re;
        double r591815 = r591814 * r591814;
        double r591816 = im;
        double r591817 = r591816 * r591816;
        double r591818 = r591815 + r591817;
        double r591819 = sqrt(r591818);
        double r591820 = r591819 - r591814;
        double r591821 = r591813 * r591820;
        double r591822 = sqrt(r591821);
        double r591823 = r591812 * r591822;
        return r591823;
}
↓
double f(double re, double im) {
        double r591824 = re;
        double r591825 = im;
        double r591826 = hypot(r591824, r591825);
        double r591827 = r591826 - r591824;
        double r591828 = 2.0;
        double r591829 = r591827 * r591828;
        double r591830 = sqrt(r591829);
        double r591831 = 0.5;
        double r591832 = r591830 * r591831;
        return r591832;
}

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 38.4

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

  2. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2019172 +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)))))