Average Error: 38.6 → 13.4
Time: 3.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \cdot im \le 1.66380688330522607 \cdot 10^{-89} \lor \neg \left(im \cdot im \le 5.3537502599579206 \cdot 10^{-60} \lor \neg \left(im \cdot im \le 1.1238092912873705 \cdot 10^{46} \lor \neg \left(im \cdot im \le 9.7188804582487216 \cdot 10^{283}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \cdot im \le 1.66380688330522607 \cdot 10^{-89} \lor \neg \left(im \cdot im \le 5.3537502599579206 \cdot 10^{-60} \lor \neg \left(im \cdot im \le 1.1238092912873705 \cdot 10^{46} \lor \neg \left(im \cdot im \le 9.7188804582487216 \cdot 10^{283}\right)\right)\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r171792 = 0.5;
        double r171793 = 2.0;
        double r171794 = re;
        double r171795 = r171794 * r171794;
        double r171796 = im;
        double r171797 = r171796 * r171796;
        double r171798 = r171795 + r171797;
        double r171799 = sqrt(r171798);
        double r171800 = r171799 + r171794;
        double r171801 = r171793 * r171800;
        double r171802 = sqrt(r171801);
        double r171803 = r171792 * r171802;
        return r171803;
}
↓
double f(double re, double im) {
        double r171804 = im;
        double r171805 = r171804 * r171804;
        double r171806 = 1.663806883305226e-89;
        bool r171807 = r171805 <= r171806;
        double r171808 = 5.353750259957921e-60;
        bool r171809 = r171805 <= r171808;
        double r171810 = 1.1238092912873705e+46;
        bool r171811 = r171805 <= r171810;
        double r171812 = 9.718880458248722e+283;
        bool r171813 = r171805 <= r171812;
        double r171814 = !r171813;
        bool r171815 = r171811 || r171814;
        double r171816 = !r171815;
        bool r171817 = r171809 || r171816;
        double r171818 = !r171817;
        bool r171819 = r171807 || r171818;
        double r171820 = 0.5;
        double r171821 = 2.0;
        double r171822 = 1.0;
        double r171823 = re;
        double r171824 = hypot(r171823, r171804);
        double r171825 = r171822 * r171824;
        double r171826 = r171825 + r171823;
        double r171827 = r171821 * r171826;
        double r171828 = sqrt(r171827);
        double r171829 = r171820 * r171828;
        double r171830 = r171824 - r171823;
        double r171831 = r171805 / r171830;
        double r171832 = r171821 * r171831;
        double r171833 = sqrt(r171832);
        double r171834 = r171820 * r171833;
        double r171835 = r171819 ? r171829 : r171834;
        return r171835;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target34.0
Herbie13.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* im im) < 1.663806883305226e-89 or 5.353750259957921e-60 < (* im im) < 1.1238092912873705e+46 or 9.718880458248722e+283 < (* im im)

    1. Initial program 43.4

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

    3. Applied *-un-lft-identity43.4

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

    4. Applied sqrt-prod43.4

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

    5. Simplified43.4

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

    6. Simplified13.7

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

    if 1.663806883305226e-89 < (* im im) < 5.353750259957921e-60 or 1.1238092912873705e+46 < (* im im) < 9.718880458248722e+283

    1. Initial program 20.9

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

    3. Applied flip-+24.7

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

    4. Simplified20.4

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

    5. Simplified12.1

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

  4. Final simplification13.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \le 1.66380688330522607 \cdot 10^{-89} \lor \neg \left(im \cdot im \le 5.3537502599579206 \cdot 10^{-60} \lor \neg \left(im \cdot im \le 1.1238092912873705 \cdot 10^{46} \lor \neg \left(im \cdot im \le 9.7188804582487216 \cdot 10^{283}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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)))))