Average Error: 38.5 → 13.2
Time: 17.8s
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 r715484 = 0.5;
        double r715485 = 2.0;
        double r715486 = re;
        double r715487 = r715486 * r715486;
        double r715488 = im;
        double r715489 = r715488 * r715488;
        double r715490 = r715487 + r715489;
        double r715491 = sqrt(r715490);
        double r715492 = r715491 - r715486;
        double r715493 = r715485 * r715492;
        double r715494 = sqrt(r715493);
        double r715495 = r715484 * r715494;
        return r715495;
}
↓
double f(double re, double im) {
        double r715496 = re;
        double r715497 = im;
        double r715498 = hypot(r715496, r715497);
        double r715499 = r715498 - r715496;
        double r715500 = 2.0;
        double r715501 = r715499 * r715500;
        double r715502 = sqrt(r715501);
        double r715503 = 0.5;
        double r715504 = r715502 * r715503;
        return r715504;
}

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 38.5

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

  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 2019192 +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)))))