Average Error: 38.9 → 13.6
Time: 20.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2} \cdot 0.5
double f(double re, double im) {
        double r6451702 = 0.5;
        double r6451703 = 2.0;
        double r6451704 = re;
        double r6451705 = r6451704 * r6451704;
        double r6451706 = im;
        double r6451707 = r6451706 * r6451706;
        double r6451708 = r6451705 + r6451707;
        double r6451709 = sqrt(r6451708);
        double r6451710 = r6451709 + r6451704;
        double r6451711 = r6451703 * r6451710;
        double r6451712 = sqrt(r6451711);
        double r6451713 = r6451702 * r6451712;
        return r6451713;
}
↓
double f(double re, double im) {
        double r6451714 = re;
        double r6451715 = im;
        double r6451716 = hypot(r6451714, r6451715);
        double r6451717 = r6451714 + r6451716;
        double r6451718 = 2.0;
        double r6451719 = r6451717 * r6451718;
        double r6451720 = sqrt(r6451719);
        double r6451721 = 0.5;
        double r6451722 = r6451720 * r6451721;
        return r6451722;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target33.7
Herbie13.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.9

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

  2. Simplified13.6

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

  3. Final simplification13.6

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (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)))))