Average Error: 39.1 → 27.9
Time: 14.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.050261451950869383577954983697308069583 \cdot 10^{-204}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 5.805113327902052729711003049635275164279 \cdot 10^{-173}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 4.779501075464754882569321337942348923064 \cdot 10^{124}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.050261451950869383577954983697308069583 \cdot 10^{-204}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 5.805113327902052729711003049635275164279 \cdot 10^{-173}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\

\mathbf{elif}\;re \le 4.779501075464754882569321337942348923064 \cdot 10^{124}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r240326 = 0.5;
        double r240327 = 2.0;
        double r240328 = re;
        double r240329 = r240328 * r240328;
        double r240330 = im;
        double r240331 = r240330 * r240330;
        double r240332 = r240329 + r240331;
        double r240333 = sqrt(r240332);
        double r240334 = r240333 + r240328;
        double r240335 = r240327 * r240334;
        double r240336 = sqrt(r240335);
        double r240337 = r240326 * r240336;
        return r240337;
}
↓
double f(double re, double im) {
        double r240338 = re;
        double r240339 = -3.0502614519508694e-204;
        bool r240340 = r240338 <= r240339;
        double r240341 = 0.5;
        double r240342 = 2.0;
        double r240343 = im;
        double r240344 = 2.0;
        double r240345 = pow(r240343, r240344);
        double r240346 = r240338 * r240338;
        double r240347 = r240343 * r240343;
        double r240348 = r240346 + r240347;
        double r240349 = sqrt(r240348);
        double r240350 = r240349 - r240338;
        double r240351 = r240345 / r240350;
        double r240352 = r240342 * r240351;
        double r240353 = sqrt(r240352);
        double r240354 = r240341 * r240353;
        double r240355 = 5.805113327902053e-173;
        bool r240356 = r240338 <= r240355;
        double r240357 = r240343 + r240338;
        double r240358 = r240342 * r240357;
        double r240359 = sqrt(r240358);
        double r240360 = r240341 * r240359;
        double r240361 = 4.779501075464755e+124;
        bool r240362 = r240338 <= r240361;
        double r240363 = sqrt(r240349);
        double r240364 = cbrt(r240348);
        double r240365 = r240364 * r240364;
        double r240366 = sqrt(r240365);
        double r240367 = sqrt(r240366);
        double r240368 = r240363 * r240367;
        double r240369 = sqrt(r240364);
        double r240370 = sqrt(r240369);
        double r240371 = r240368 * r240370;
        double r240372 = r240371 + r240338;
        double r240373 = r240342 * r240372;
        double r240374 = sqrt(r240373);
        double r240375 = r240341 * r240374;
        double r240376 = r240344 * r240338;
        double r240377 = r240342 * r240376;
        double r240378 = sqrt(r240377);
        double r240379 = r240341 * r240378;
        double r240380 = r240362 ? r240375 : r240379;
        double r240381 = r240356 ? r240360 : r240380;
        double r240382 = r240340 ? r240354 : r240381;
        return r240382;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.1
Target34.3
Herbie27.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -3.0502614519508694e-204

    1. Initial program 49.2

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

    3. Applied flip-+49.2

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

    4. Simplified37.9

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

    if -3.0502614519508694e-204 < re < 5.805113327902053e-173

    1. Initial program 30.6

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

    2. Taylor expanded around 0 34.1

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

    if 5.805113327902053e-173 < re < 4.779501075464755e+124

    1. Initial program 16.8

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

    3. Applied add-sqr-sqrt16.8

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

    4. Applied sqrt-prod16.9

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

    6. Applied add-cube-cbrt17.0

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

    7. Applied sqrt-prod17.0

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

    8. Applied sqrt-prod17.0

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

    9. Applied associate-*r*17.0

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

    if 4.779501075464755e+124 < re

    1. Initial program 57.1

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

    2. Taylor expanded around inf 8.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification27.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.050261451950869383577954983697308069583 \cdot 10^{-204}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 5.805113327902052729711003049635275164279 \cdot 10^{-173}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 4.779501075464754882569321337942348923064 \cdot 10^{124}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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)))))