Average Error: 38.5 → 13.5
Time: 21.4s
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 r110497 = 0.5;
        double r110498 = 2.0;
        double r110499 = re;
        double r110500 = r110499 * r110499;
        double r110501 = im;
        double r110502 = r110501 * r110501;
        double r110503 = r110500 + r110502;
        double r110504 = sqrt(r110503);
        double r110505 = r110504 + r110499;
        double r110506 = r110498 * r110505;
        double r110507 = sqrt(r110506);
        double r110508 = r110497 * r110507;
        return r110508;
}
↓
double f(double re, double im) {
        double r110509 = 0.5;
        double r110510 = re;
        double r110511 = im;
        double r110512 = hypot(r110510, r110511);
        double r110513 = r110510 + r110512;
        double r110514 = 2.0;
        double r110515 = r110513 * r110514;
        double r110516 = sqrt(r110515);
        double r110517 = r110509 * r110516;
        return r110517;
}

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)))))