Average Error: 38.7 → 26.2
Time: 5.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.77145783633808892 \cdot 10^{-294}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 5.27522725235634779 \cdot 10^{-220}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 1.20526441884369862 \cdot 10^{151}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.77145783633808892 \cdot 10^{-294}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 5.27522725235634779 \cdot 10^{-220}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \le 1.20526441884369862 \cdot 10^{151}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{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 r360300 = 0.5;
        double r360301 = 2.0;
        double r360302 = re;
        double r360303 = r360302 * r360302;
        double r360304 = im;
        double r360305 = r360304 * r360304;
        double r360306 = r360303 + r360305;
        double r360307 = sqrt(r360306);
        double r360308 = r360307 + r360302;
        double r360309 = r360301 * r360308;
        double r360310 = sqrt(r360309);
        double r360311 = r360300 * r360310;
        return r360311;
}
↓
double f(double re, double im) {
        double r360312 = re;
        double r360313 = -3.771457836338089e-294;
        bool r360314 = r360312 <= r360313;
        double r360315 = 0.5;
        double r360316 = 2.0;
        double r360317 = im;
        double r360318 = 2.0;
        double r360319 = pow(r360317, r360318);
        double r360320 = r360312 * r360312;
        double r360321 = r360317 * r360317;
        double r360322 = r360320 + r360321;
        double r360323 = sqrt(r360322);
        double r360324 = r360323 - r360312;
        double r360325 = r360319 / r360324;
        double r360326 = r360316 * r360325;
        double r360327 = sqrt(r360326);
        double r360328 = r360315 * r360327;
        double r360329 = 5.275227252356348e-220;
        bool r360330 = r360312 <= r360329;
        double r360331 = r360312 + r360317;
        double r360332 = r360316 * r360331;
        double r360333 = sqrt(r360332);
        double r360334 = r360315 * r360333;
        double r360335 = 1.2052644188436986e+151;
        bool r360336 = r360312 <= r360335;
        double r360337 = r360323 + r360312;
        double r360338 = r360316 * r360337;
        double r360339 = sqrt(r360338);
        double r360340 = r360315 * r360339;
        double r360341 = r360318 * r360312;
        double r360342 = r360316 * r360341;
        double r360343 = sqrt(r360342);
        double r360344 = r360315 * r360343;
        double r360345 = r360336 ? r360340 : r360344;
        double r360346 = r360330 ? r360334 : r360345;
        double r360347 = r360314 ? r360328 : r360346;
        return r360347;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.7
Target33.1
Herbie26.2
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -3.771457836338089e-294

    1. Initial program 46.6

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

    3. Applied flip-+46.5

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

    4. Simplified35.1

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

    if -3.771457836338089e-294 < re < 5.275227252356348e-220

    1. Initial program 30.4

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

    2. Taylor expanded around 0 32.0

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

    if 5.275227252356348e-220 < re < 1.2052644188436986e+151

    1. Initial program 17.7

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

    if 1.2052644188436986e+151 < re

    1. Initial program 63.0

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

    2. Taylor expanded around inf 7.4

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

  4. Final simplification26.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.77145783633808892 \cdot 10^{-294}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 5.27522725235634779 \cdot 10^{-220}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 1.20526441884369862 \cdot 10^{151}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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)))))