Average Error: 37.2 → 12.9
Time: 15.5s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r7165601 = 0.5;
        double r7165602 = 2.0;
        double r7165603 = re;
        double r7165604 = r7165603 * r7165603;
        double r7165605 = im;
        double r7165606 = r7165605 * r7165605;
        double r7165607 = r7165604 + r7165606;
        double r7165608 = sqrt(r7165607);
        double r7165609 = r7165608 + r7165603;
        double r7165610 = r7165602 * r7165609;
        double r7165611 = sqrt(r7165610);
        double r7165612 = r7165601 * r7165611;
        return r7165612;
}
↓
double f(double re, double im) {
        double r7165613 = re;
        double r7165614 = im;
        double r7165615 = hypot(r7165613, r7165614);
        double r7165616 = r7165613 + r7165615;
        double r7165617 = 2.0;
        double r7165618 = r7165616 * r7165617;
        double r7165619 = sqrt(r7165618);
        double r7165620 = 0.5;
        double r7165621 = r7165619 * r7165620;
        return r7165621;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target32.1
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.2

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

  2. Simplified12.9

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

  3. Final simplification12.9

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))