Average Error: 37.7 → 13.3
Time: 24.2s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r598574 = 0.5;
        double r598575 = 2.0;
        double r598576 = re;
        double r598577 = r598576 * r598576;
        double r598578 = im;
        double r598579 = r598578 * r598578;
        double r598580 = r598577 + r598579;
        double r598581 = sqrt(r598580);
        double r598582 = r598581 - r598576;
        double r598583 = r598575 * r598582;
        double r598584 = sqrt(r598583);
        double r598585 = r598574 * r598584;
        return r598585;
}
↓
double f(double re, double im) {
        double r598586 = re;
        double r598587 = im;
        double r598588 = hypot(r598586, r598587);
        double r598589 = r598588 - r598586;
        double r598590 = 2.0;
        double r598591 = r598589 * r598590;
        double r598592 = sqrt(r598591);
        double r598593 = 0.5;
        double r598594 = r598592 * r598593;
        return r598594;
}

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 37.7

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

  2. Simplified13.3

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0}}\]

  3. Final simplification13.3

    \[\leadsto \sqrt{\left(\sqrt{re^2 + im^2}^* - re\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019104 +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)))))