Average Error: 38.8 → 24.3
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.34738501180888645 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-1 \cdot re - re}}\\ \mathbf{elif}\;re \le -7.4245329977218585 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le -3.3124292600034948 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.41793787035234448 \cdot 10^{-242}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{im}}}\\ \mathbf{elif}\;re \le 4.08844438160230292 \cdot 10^{68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.34738501180888645 \cdot 10^{154}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-1 \cdot re - re}}\\

\mathbf{elif}\;re \le -7.4245329977218585 \cdot 10^{-93}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le -3.3124292600034948 \cdot 10^{-125}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\

\mathbf{elif}\;re \le 1.41793787035234448 \cdot 10^{-242}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{im}}}\\

\mathbf{elif}\;re \le 4.08844438160230292 \cdot 10^{68}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r121292 = 0.5;
        double r121293 = 2.0;
        double r121294 = re;
        double r121295 = r121294 * r121294;
        double r121296 = im;
        double r121297 = r121296 * r121296;
        double r121298 = r121295 + r121297;
        double r121299 = sqrt(r121298);
        double r121300 = r121299 + r121294;
        double r121301 = r121293 * r121300;
        double r121302 = sqrt(r121301);
        double r121303 = r121292 * r121302;
        return r121303;
}
↓
double f(double re, double im) {
        double r121304 = re;
        double r121305 = -1.3473850118088864e+154;
        bool r121306 = r121304 <= r121305;
        double r121307 = 0.5;
        double r121308 = 2.0;
        double r121309 = im;
        double r121310 = r121309 * r121309;
        double r121311 = -1.0;
        double r121312 = r121311 * r121304;
        double r121313 = r121312 - r121304;
        double r121314 = r121310 / r121313;
        double r121315 = r121308 * r121314;
        double r121316 = sqrt(r121315);
        double r121317 = r121307 * r121316;
        double r121318 = -7.4245329977218585e-93;
        bool r121319 = r121304 <= r121318;
        double r121320 = r121308 * r121310;
        double r121321 = sqrt(r121320);
        double r121322 = r121304 * r121304;
        double r121323 = r121322 + r121310;
        double r121324 = sqrt(r121323);
        double r121325 = r121324 - r121304;
        double r121326 = sqrt(r121325);
        double r121327 = r121321 / r121326;
        double r121328 = r121307 * r121327;
        double r121329 = -3.312429260003495e-125;
        bool r121330 = r121304 <= r121329;
        double r121331 = r121309 + r121304;
        double r121332 = r121308 * r121331;
        double r121333 = sqrt(r121332);
        double r121334 = r121307 * r121333;
        double r121335 = 1.4179378703523445e-242;
        bool r121336 = r121304 <= r121335;
        double r121337 = r121325 / r121309;
        double r121338 = r121309 / r121337;
        double r121339 = r121308 * r121338;
        double r121340 = sqrt(r121339);
        double r121341 = r121307 * r121340;
        double r121342 = 4.088444381602303e+68;
        bool r121343 = r121304 <= r121342;
        double r121344 = 0.5;
        double r121345 = pow(r121323, r121344);
        double r121346 = r121345 + r121304;
        double r121347 = r121308 * r121346;
        double r121348 = sqrt(r121347);
        double r121349 = r121307 * r121348;
        double r121350 = 2.0;
        double r121351 = r121350 * r121304;
        double r121352 = r121308 * r121351;
        double r121353 = sqrt(r121352);
        double r121354 = r121307 * r121353;
        double r121355 = r121343 ? r121349 : r121354;
        double r121356 = r121336 ? r121341 : r121355;
        double r121357 = r121330 ? r121334 : r121356;
        double r121358 = r121319 ? r121328 : r121357;
        double r121359 = r121306 ? r121317 : r121358;
        return r121359;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.8
Target34.2
Herbie24.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 6 regimes

  2. if re < -1.3473850118088864e+154

    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. Simplified51.6

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

    5. Taylor expanded around -inf 32.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\color{blue}{-1 \cdot re} - re}}\]

    if -1.3473850118088864e+154 < re < -7.4245329977218585e-93

    1. Initial program 47.6

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

    3. Applied flip-+47.6

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

    4. Simplified32.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied associate-*r/32.7

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

    7. Applied sqrt-div30.6

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

    if -7.4245329977218585e-93 < re < -3.312429260003495e-125

    1. Initial program 32.9

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

    2. Taylor expanded around 0 39.9

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

    if -3.312429260003495e-125 < re < 1.4179378703523445e-242

    1. Initial program 31.5

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

    3. Applied flip-+31.4

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

    4. Simplified31.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied associate-/l*29.2

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

    if 1.4179378703523445e-242 < re < 4.088444381602303e+68

    1. Initial program 19.3

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

    3. Applied add-sqr-sqrt19.3

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

    4. Applied sqrt-prod19.4

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

    6. Applied pow119.4

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

    7. Applied sqrt-pow119.4

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

    8. Applied pow119.4

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

    9. Applied sqrt-pow119.4

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

    10. Applied pow-prod-down19.3

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

    11. Simplified19.3

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

    if 4.088444381602303e+68 < re

    1. Initial program 47.9

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

    2. Taylor expanded around inf 12.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
  3. Recombined 6 regimes into one program.

  4. Final simplification24.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.34738501180888645 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-1 \cdot re - re}}\\ \mathbf{elif}\;re \le -7.4245329977218585 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le -3.3124292600034948 \cdot 10^{-125}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.41793787035234448 \cdot 10^{-242}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{im}}}\\ \mathbf{elif}\;re \le 4.08844438160230292 \cdot 10^{68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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)))))