Average Error: 37.4 → 12.7
Time: 14.1s
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 r309585 = 0.5;
        double r309586 = 2.0;
        double r309587 = re;
        double r309588 = r309587 * r309587;
        double r309589 = im;
        double r309590 = r309589 * r309589;
        double r309591 = r309588 + r309590;
        double r309592 = sqrt(r309591);
        double r309593 = r309592 - r309587;
        double r309594 = r309586 * r309593;
        double r309595 = sqrt(r309594);
        double r309596 = r309585 * r309595;
        return r309596;
}
↓
double f(double re, double im) {
        double r309597 = re;
        double r309598 = im;
        double r309599 = hypot(r309597, r309598);
        double r309600 = r309599 - r309597;
        double r309601 = 2.0;
        double r309602 = r309600 * r309601;
        double r309603 = sqrt(r309602);
        double r309604 = 0.5;
        double r309605 = r309603 * r309604;
        return r309605;
}

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 37.4

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

  2. Simplified12.7

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

  3. Final simplification12.7

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

Reproduce

herbie shell --seed 2019153 +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)))))