Average Error: 38.3 → 12.0
Time: 3.3s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -17761354360071876897040102379133383737340 \lor \neg \left(re \le -1.013620097950354583113738828406558134532 \cdot 10^{-7} \lor \neg \left(re \le -1.941232155667051907635267844609624530636 \cdot 10^{-71}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \sqrt{1} \cdot \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -17761354360071876897040102379133383737340 \lor \neg \left(re \le -1.013620097950354583113738828406558134532 \cdot 10^{-7} \lor \neg \left(re \le -1.941232155667051907635267844609624530636 \cdot 10^{-71}\right)\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \sqrt{1} \cdot \mathsf{hypot}\left(re, im\right)\right)\right)}\\

\end{array}
double f(double re, double im) {
        double r192688 = 0.5;
        double r192689 = 2.0;
        double r192690 = re;
        double r192691 = r192690 * r192690;
        double r192692 = im;
        double r192693 = r192692 * r192692;
        double r192694 = r192691 + r192693;
        double r192695 = sqrt(r192694);
        double r192696 = r192695 + r192690;
        double r192697 = r192689 * r192696;
        double r192698 = sqrt(r192697);
        double r192699 = r192688 * r192698;
        return r192699;
}
↓
double f(double re, double im) {
        double r192700 = re;
        double r192701 = -1.7761354360071877e+40;
        bool r192702 = r192700 <= r192701;
        double r192703 = -1.0136200979503546e-07;
        bool r192704 = r192700 <= r192703;
        double r192705 = -1.941232155667052e-71;
        bool r192706 = r192700 <= r192705;
        double r192707 = !r192706;
        bool r192708 = r192704 || r192707;
        double r192709 = !r192708;
        bool r192710 = r192702 || r192709;
        double r192711 = 0.5;
        double r192712 = 2.0;
        double r192713 = im;
        double r192714 = 2.0;
        double r192715 = pow(r192713, r192714);
        double r192716 = hypot(r192700, r192713);
        double r192717 = r192716 - r192700;
        double r192718 = r192715 / r192717;
        double r192719 = r192712 * r192718;
        double r192720 = sqrt(r192719);
        double r192721 = r192711 * r192720;
        double r192722 = 1.0;
        double r192723 = sqrt(r192722);
        double r192724 = r192723 * r192716;
        double r192725 = r192700 + r192724;
        double r192726 = r192722 * r192725;
        double r192727 = r192712 * r192726;
        double r192728 = sqrt(r192727);
        double r192729 = r192711 * r192728;
        double r192730 = r192710 ? r192721 : r192729;
        return r192730;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.3
Target33.4
Herbie12.0
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -1.7761354360071877e+40 or -1.0136200979503546e-07 < re < -1.941232155667052e-71

    1. Initial program 54.9

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

    3. Applied flip-+54.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Simplified40.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]

    5. Simplified31.7

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

    if -1.7761354360071877e+40 < re < -1.0136200979503546e-07 or -1.941232155667052e-71 < re

    1. Initial program 32.1

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

    3. Applied *-un-lft-identity32.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + \color{blue}{1 \cdot re}\right)}\]

    4. Applied *-un-lft-identity32.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \sqrt{re \cdot re + im \cdot im}} + 1 \cdot re\right)}\]

    5. Applied distribute-lft-out32.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)}}\]

    6. Simplified4.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt4.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \color{blue}{\sqrt{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}}\right)\right)}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity4.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \sqrt{\color{blue}{1 \cdot \mathsf{hypot}\left(re, im\right)}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}\right)\right)}\]

    11. Applied sqrt-prod4.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \color{blue}{\left(\sqrt{1} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}\right)\right)}\]

    12. Applied associate-*l*4.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \color{blue}{\sqrt{1} \cdot \left(\sqrt{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}\right)}\right)\right)}\]

    13. Simplified4.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \sqrt{1} \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -17761354360071876897040102379133383737340 \lor \neg \left(re \le -1.013620097950354583113738828406558134532 \cdot 10^{-7} \lor \neg \left(re \le -1.941232155667051907635267844609624530636 \cdot 10^{-71}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \sqrt{1} \cdot \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))))