Average Error: 37.2 → 13.2
Time: 27.4s
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 r842496 = 0.5;
        double r842497 = 2.0;
        double r842498 = re;
        double r842499 = r842498 * r842498;
        double r842500 = im;
        double r842501 = r842500 * r842500;
        double r842502 = r842499 + r842501;
        double r842503 = sqrt(r842502);
        double r842504 = r842503 - r842498;
        double r842505 = r842497 * r842504;
        double r842506 = sqrt(r842505);
        double r842507 = r842496 * r842506;
        return r842507;
}
↓
double f(double re, double im) {
        double r842508 = re;
        double r842509 = im;
        double r842510 = hypot(r842508, r842509);
        double r842511 = r842510 - r842508;
        double r842512 = 2.0;
        double r842513 = r842511 * r842512;
        double r842514 = sqrt(r842513);
        double r842515 = 0.5;
        double r842516 = r842514 * r842515;
        return r842516;
}

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 37.2

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

  2. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2019138 +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)))))