Average Error: 37.8 → 13.5
Time: 14.4s
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 r5613546 = 0.5;
        double r5613547 = 2.0;
        double r5613548 = re;
        double r5613549 = r5613548 * r5613548;
        double r5613550 = im;
        double r5613551 = r5613550 * r5613550;
        double r5613552 = r5613549 + r5613551;
        double r5613553 = sqrt(r5613552);
        double r5613554 = r5613553 + r5613548;
        double r5613555 = r5613547 * r5613554;
        double r5613556 = sqrt(r5613555);
        double r5613557 = r5613546 * r5613556;
        return r5613557;
}
↓
double f(double re, double im) {
        double r5613558 = re;
        double r5613559 = im;
        double r5613560 = hypot(r5613558, r5613559);
        double r5613561 = r5613558 + r5613560;
        double r5613562 = 2.0;
        double r5613563 = r5613561 * r5613562;
        double r5613564 = sqrt(r5613563);
        double r5613565 = 0.5;
        double r5613566 = r5613564 * r5613565;
        return r5613566;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.8
Target33.0
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.8

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

  2. Simplified13.5

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

  3. Final simplification13.5

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

Reproduce

herbie shell --seed 2019146 +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)))))