Average Error: 38.3 → 12.0
Time: 3.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -17761354360071876897040102379133383737340 \lor \neg \left(re \le -1.013620097950354583113738828406558134532 \cdot 10^{-7} \lor \neg \left(re \le -1.941232155667051907635267844609624530636 \cdot 10^{-71}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
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 \mathsf{hypot}\left(re, im\right) + re\right)}\\

\end{array}
double f(double re, double im) {
        double r252395 = 0.5;
        double r252396 = 2.0;
        double r252397 = re;
        double r252398 = r252397 * r252397;
        double r252399 = im;
        double r252400 = r252399 * r252399;
        double r252401 = r252398 + r252400;
        double r252402 = sqrt(r252401);
        double r252403 = r252402 + r252397;
        double r252404 = r252396 * r252403;
        double r252405 = sqrt(r252404);
        double r252406 = r252395 * r252405;
        return r252406;
}
↓
double f(double re, double im) {
        double r252407 = re;
        double r252408 = -1.7761354360071877e+40;
        bool r252409 = r252407 <= r252408;
        double r252410 = -1.0136200979503546e-07;
        bool r252411 = r252407 <= r252410;
        double r252412 = -1.941232155667052e-71;
        bool r252413 = r252407 <= r252412;
        double r252414 = !r252413;
        bool r252415 = r252411 || r252414;
        double r252416 = !r252415;
        bool r252417 = r252409 || r252416;
        double r252418 = 0.5;
        double r252419 = 2.0;
        double r252420 = im;
        double r252421 = 2.0;
        double r252422 = pow(r252420, r252421);
        double r252423 = hypot(r252407, r252420);
        double r252424 = r252423 - r252407;
        double r252425 = r252422 / r252424;
        double r252426 = r252419 * r252425;
        double r252427 = sqrt(r252426);
        double r252428 = r252418 * r252427;
        double r252429 = 1.0;
        double r252430 = r252429 * r252423;
        double r252431 = r252430 + r252407;
        double r252432 = r252419 * r252431;
        double r252433 = sqrt(r252432);
        double r252434 = r252418 * r252433;
        double r252435 = r252417 ? r252428 : r252434;
        return r252435;
}

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

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

    4. Applied sqrt-prod32.1

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

    5. Simplified32.1

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

    6. Simplified4.6

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

  4. Final simplification12.0

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

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)))))