Average Error: 39.5 → 19.7
Time: 9.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.971834220295259 \cdot 10^{153}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{{im}^{2} \cdot 2}}{\sqrt{-2 \cdot re}}\\ \mathbf{elif}\;re \le -1.2504367945899628 \cdot 10^{-181}:\\ \;\;\;\;0.5 \cdot \left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)\\ \mathbf{elif}\;re \le 1.15471890189012987 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.3861488470850941 \cdot 10^{97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \left(\sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.971834220295259 \cdot 10^{153}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{{im}^{2} \cdot 2}}{\sqrt{-2 \cdot re}}\\

\mathbf{elif}\;re \le -1.2504367945899628 \cdot 10^{-181}:\\
\;\;\;\;0.5 \cdot \left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)\\

\mathbf{elif}\;re \le 1.15471890189012987 \cdot 10^{-253}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\

\mathbf{elif}\;re \le 1.3861488470850941 \cdot 10^{97}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \left(\sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right)} + re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\

\end{array}
double f(double re, double im) {
        double r319287 = 0.5;
        double r319288 = 2.0;
        double r319289 = re;
        double r319290 = r319289 * r319289;
        double r319291 = im;
        double r319292 = r319291 * r319291;
        double r319293 = r319290 + r319292;
        double r319294 = sqrt(r319293);
        double r319295 = r319294 + r319289;
        double r319296 = r319288 * r319295;
        double r319297 = sqrt(r319296);
        double r319298 = r319287 * r319297;
        return r319298;
}
↓
double f(double re, double im) {
        double r319299 = re;
        double r319300 = -1.971834220295259e+153;
        bool r319301 = r319299 <= r319300;
        double r319302 = 0.5;
        double r319303 = im;
        double r319304 = 2.0;
        double r319305 = pow(r319303, r319304);
        double r319306 = 2.0;
        double r319307 = r319305 * r319306;
        double r319308 = sqrt(r319307);
        double r319309 = -2.0;
        double r319310 = r319309 * r319299;
        double r319311 = sqrt(r319310);
        double r319312 = r319308 / r319311;
        double r319313 = r319302 * r319312;
        double r319314 = -1.2504367945899628e-181;
        bool r319315 = r319299 <= r319314;
        double r319316 = sqrt(r319306);
        double r319317 = r319299 * r319299;
        double r319318 = r319303 * r319303;
        double r319319 = r319317 + r319318;
        double r319320 = sqrt(r319319);
        double r319321 = r319320 - r319299;
        double r319322 = sqrt(r319321);
        double r319323 = sqrt(r319322);
        double r319324 = r319316 / r319323;
        double r319325 = fabs(r319303);
        double r319326 = r319325 / r319323;
        double r319327 = r319324 * r319326;
        double r319328 = r319302 * r319327;
        double r319329 = 1.1547189018901299e-253;
        bool r319330 = r319299 <= r319329;
        double r319331 = r319303 + r319299;
        double r319332 = r319306 * r319331;
        double r319333 = sqrt(r319332);
        double r319334 = r319302 * r319333;
        double r319335 = 1.386148847085094e+97;
        bool r319336 = r319299 <= r319335;
        double r319337 = cbrt(r319319);
        double r319338 = r319337 * r319337;
        double r319339 = cbrt(r319338);
        double r319340 = cbrt(r319337);
        double r319341 = r319339 * r319340;
        double r319342 = r319338 * r319341;
        double r319343 = sqrt(r319342);
        double r319344 = r319343 + r319299;
        double r319345 = r319306 * r319344;
        double r319346 = sqrt(r319345);
        double r319347 = r319302 * r319346;
        double r319348 = r319299 + r319299;
        double r319349 = r319306 * r319348;
        double r319350 = sqrt(r319349);
        double r319351 = r319302 * r319350;
        double r319352 = r319336 ? r319347 : r319351;
        double r319353 = r319330 ? r319334 : r319352;
        double r319354 = r319315 ? r319328 : r319353;
        double r319355 = r319301 ? r319313 : r319354;
        return r319355;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.5
Target34.4
Herbie19.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -1.971834220295259e+153

    1. Initial program 64.0

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

    3. Applied flip-+64.0

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

    4. Applied associate-*r/64.0

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

    5. Applied sqrt-div64.0

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

    6. Simplified49.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left({im}^{2} + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    7. Taylor expanded around -inf 19.1

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left({im}^{2} + 0\right)}}{\sqrt{\color{blue}{-2 \cdot re}}}\]

    if -1.971834220295259e+153 < re < -1.2504367945899628e-181

    1. Initial program 44.1

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

    3. Applied flip-+44.0

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

    4. Applied associate-*r/44.1

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

    5. Applied sqrt-div44.1

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

    6. Simplified30.2

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left({im}^{2} + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt30.2

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left({im}^{2} + 0\right)}}{\sqrt{\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}\]

    9. Applied sqrt-prod30.3

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left({im}^{2} + 0\right)}}{\color{blue}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}\]

    10. Applied sqrt-prod30.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{{im}^{2} + 0}}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    11. Applied times-frac30.3

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\sqrt{{im}^{2} + 0}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)}\]

    12. Simplified17.8

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

    if -1.2504367945899628e-181 < re < 1.1547189018901299e-253

    1. Initial program 32.7

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

    3. Applied add-cube-cbrt33.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\]

    4. Taylor expanded around 0 33.9

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

    if 1.1547189018901299e-253 < re < 1.386148847085094e+97

    1. Initial program 20.3

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

    3. Applied add-cube-cbrt20.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt20.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

    6. Applied cbrt-prod20.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right)}} + re\right)}\]

    if 1.386148847085094e+97 < re

    1. Initial program 51.2

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

    2. Taylor expanded around inf 10.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification19.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.971834220295259 \cdot 10^{153}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{{im}^{2} \cdot 2}}{\sqrt{-2 \cdot re}}\\ \mathbf{elif}\;re \le -1.2504367945899628 \cdot 10^{-181}:\\ \;\;\;\;0.5 \cdot \left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)\\ \mathbf{elif}\;re \le 1.15471890189012987 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.3861488470850941 \cdot 10^{97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \left(\sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))))