Average Error: 38.5 → 13.5
Time: 22.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}
double f(double re, double im) {
        double r168321 = 0.5;
        double r168322 = 2.0;
        double r168323 = re;
        double r168324 = r168323 * r168323;
        double r168325 = im;
        double r168326 = r168325 * r168325;
        double r168327 = r168324 + r168326;
        double r168328 = sqrt(r168327);
        double r168329 = r168328 + r168323;
        double r168330 = r168322 * r168329;
        double r168331 = sqrt(r168330);
        double r168332 = r168321 * r168331;
        return r168332;
}
↓
double f(double re, double im) {
        double r168333 = 0.5;
        double r168334 = re;
        double r168335 = im;
        double r168336 = hypot(r168334, r168335);
        double r168337 = r168334 + r168336;
        double r168338 = 2.0;
        double r168339 = r168337 * r168338;
        double r168340 = sqrt(r168339);
        double r168341 = r168333 * r168340;
        return r168341;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.5
Target33.3
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.5

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

  2. Simplified13.5

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

  3. Final simplification13.5

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))