Average Error: 37.6 → 13.3
Time: 13.3s
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 r24662776 = 0.5;
        double r24662777 = 2.0;
        double r24662778 = re;
        double r24662779 = r24662778 * r24662778;
        double r24662780 = im;
        double r24662781 = r24662780 * r24662780;
        double r24662782 = r24662779 + r24662781;
        double r24662783 = sqrt(r24662782);
        double r24662784 = r24662783 + r24662778;
        double r24662785 = r24662777 * r24662784;
        double r24662786 = sqrt(r24662785);
        double r24662787 = r24662776 * r24662786;
        return r24662787;
}
↓
double f(double re, double im) {
        double r24662788 = re;
        double r24662789 = im;
        double r24662790 = hypot(r24662788, r24662789);
        double r24662791 = r24662788 + r24662790;
        double r24662792 = 2.0;
        double r24662793 = r24662791 * r24662792;
        double r24662794 = sqrt(r24662793);
        double r24662795 = 0.5;
        double r24662796 = r24662794 * r24662795;
        return r24662796;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target32.6
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.6

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

  2. Simplified13.3

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

  3. Final simplification13.3

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

Reproduce

herbie shell --seed 2019107 +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)))))