Average Error: 38.8 → 13.1
Time: 20.0s
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 r902794 = 0.5;
        double r902795 = 2.0;
        double r902796 = re;
        double r902797 = r902796 * r902796;
        double r902798 = im;
        double r902799 = r902798 * r902798;
        double r902800 = r902797 + r902799;
        double r902801 = sqrt(r902800);
        double r902802 = r902801 - r902796;
        double r902803 = r902795 * r902802;
        double r902804 = sqrt(r902803);
        double r902805 = r902794 * r902804;
        return r902805;
}
↓
double f(double re, double im) {
        double r902806 = re;
        double r902807 = im;
        double r902808 = hypot(r902806, r902807);
        double r902809 = r902808 - r902806;
        double r902810 = 2.0;
        double r902811 = r902809 * r902810;
        double r902812 = sqrt(r902811);
        double r902813 = 0.5;
        double r902814 = r902812 * r902813;
        return r902814;
}

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 38.8

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019170 +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)))))