Average Error: 38.0 → 13.8
Time: 14.9s
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 r188685 = 0.5;
        double r188686 = 2.0;
        double r188687 = re;
        double r188688 = r188687 * r188687;
        double r188689 = im;
        double r188690 = r188689 * r188689;
        double r188691 = r188688 + r188690;
        double r188692 = sqrt(r188691);
        double r188693 = r188692 - r188687;
        double r188694 = r188686 * r188693;
        double r188695 = sqrt(r188694);
        double r188696 = r188685 * r188695;
        return r188696;
}
↓
double f(double re, double im) {
        double r188697 = re;
        double r188698 = im;
        double r188699 = hypot(r188697, r188698);
        double r188700 = r188699 - r188697;
        double r188701 = 2.0;
        double r188702 = r188700 * r188701;
        double r188703 = sqrt(r188702);
        double r188704 = 0.5;
        double r188705 = r188703 * r188704;
        return r188705;
}

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 38.0

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

  2. Simplified13.8

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

  3. Final simplification13.8

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

Reproduce

herbie shell --seed 2019120 +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)))))