Average Error: 38.2 → 13.5
Time: 16.8s
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 r29573 = 0.5;
        double r29574 = 2.0;
        double r29575 = re;
        double r29576 = r29575 * r29575;
        double r29577 = im;
        double r29578 = r29577 * r29577;
        double r29579 = r29576 + r29578;
        double r29580 = sqrt(r29579);
        double r29581 = r29580 - r29575;
        double r29582 = r29574 * r29581;
        double r29583 = sqrt(r29582);
        double r29584 = r29573 * r29583;
        return r29584;
}
↓
double f(double re, double im) {
        double r29585 = re;
        double r29586 = im;
        double r29587 = hypot(r29585, r29586);
        double r29588 = r29587 - r29585;
        double r29589 = 2.0;
        double r29590 = r29588 * r29589;
        double r29591 = sqrt(r29590);
        double r29592 = 0.5;
        double r29593 = r29591 * r29592;
        return r29593;
}

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 38.2

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

  2. Simplified13.5

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}}\]

  3. Final simplification13.5

    \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\]

Reproduce

herbie shell --seed 2019194 +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)))))