Average Error: 37.5 → 13.1
Time: 13.9s
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 r299649 = 0.5;
        double r299650 = 2.0;
        double r299651 = re;
        double r299652 = r299651 * r299651;
        double r299653 = im;
        double r299654 = r299653 * r299653;
        double r299655 = r299652 + r299654;
        double r299656 = sqrt(r299655);
        double r299657 = r299656 - r299651;
        double r299658 = r299650 * r299657;
        double r299659 = sqrt(r299658);
        double r299660 = r299649 * r299659;
        return r299660;
}
↓
double f(double re, double im) {
        double r299661 = re;
        double r299662 = im;
        double r299663 = hypot(r299661, r299662);
        double r299664 = r299663 - r299661;
        double r299665 = 2.0;
        double r299666 = r299664 * r299665;
        double r299667 = sqrt(r299666);
        double r299668 = 0.5;
        double r299669 = r299667 * r299668;
        return r299669;
}

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 37.5

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019154 +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)))))