Average Error: 37.3 → 13.1
Time: 20.5s
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 r885506 = 0.5;
        double r885507 = 2.0;
        double r885508 = re;
        double r885509 = r885508 * r885508;
        double r885510 = im;
        double r885511 = r885510 * r885510;
        double r885512 = r885509 + r885511;
        double r885513 = sqrt(r885512);
        double r885514 = r885513 - r885508;
        double r885515 = r885507 * r885514;
        double r885516 = sqrt(r885515);
        double r885517 = r885506 * r885516;
        return r885517;
}
↓
double f(double re, double im) {
        double r885518 = re;
        double r885519 = im;
        double r885520 = hypot(r885518, r885519);
        double r885521 = r885520 - r885518;
        double r885522 = 2.0;
        double r885523 = r885521 * r885522;
        double r885524 = sqrt(r885523);
        double r885525 = 0.5;
        double r885526 = r885524 * r885525;
        return r885526;
}

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 37.3

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

  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 2019164 +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)))))