Average Error: 38.8 → 26.4
Time: 7.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 2.409964474129751 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 5.23044770692996771 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 2.0749550549005391 \cdot 10^{149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 2.409964474129751 \cdot 10^{-295}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 5.23044770692996771 \cdot 10^{-253}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \le 2.0749550549005391 \cdot 10^{149}:\\
\;\;\;\;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 r362237 = 0.5;
        double r362238 = 2.0;
        double r362239 = re;
        double r362240 = r362239 * r362239;
        double r362241 = im;
        double r362242 = r362241 * r362241;
        double r362243 = r362240 + r362242;
        double r362244 = sqrt(r362243);
        double r362245 = r362244 + r362239;
        double r362246 = r362238 * r362245;
        double r362247 = sqrt(r362246);
        double r362248 = r362237 * r362247;
        return r362248;
}
↓
double f(double re, double im) {
        double r362249 = re;
        double r362250 = 2.409964474129751e-295;
        bool r362251 = r362249 <= r362250;
        double r362252 = 0.5;
        double r362253 = 2.0;
        double r362254 = im;
        double r362255 = 2.0;
        double r362256 = pow(r362254, r362255);
        double r362257 = r362249 * r362249;
        double r362258 = r362254 * r362254;
        double r362259 = r362257 + r362258;
        double r362260 = sqrt(r362259);
        double r362261 = r362260 - r362249;
        double r362262 = r362256 / r362261;
        double r362263 = r362253 * r362262;
        double r362264 = sqrt(r362263);
        double r362265 = r362252 * r362264;
        double r362266 = 5.230447706929968e-253;
        bool r362267 = r362249 <= r362266;
        double r362268 = r362249 + r362254;
        double r362269 = r362253 * r362268;
        double r362270 = sqrt(r362269);
        double r362271 = r362252 * r362270;
        double r362272 = 2.074955054900539e+149;
        bool r362273 = r362249 <= r362272;
        double r362274 = r362260 + r362249;
        double r362275 = r362253 * r362274;
        double r362276 = sqrt(r362275);
        double r362277 = r362252 * r362276;
        double r362278 = r362255 * r362249;
        double r362279 = r362253 * r362278;
        double r362280 = sqrt(r362279);
        double r362281 = r362252 * r362280;
        double r362282 = r362273 ? r362277 : r362281;
        double r362283 = r362267 ? r362271 : r362282;
        double r362284 = r362251 ? r362265 : r362283;
        return r362284;
}

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
Target33.7
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < 2.409964474129751e-295

    1. Initial program 45.7

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

    3. Applied flip-+45.6

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

    4. Simplified35.5

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

    if 2.409964474129751e-295 < re < 5.230447706929968e-253

    1. Initial program 32.0

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

    2. Taylor expanded around 0 32.2

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

    if 5.230447706929968e-253 < re < 2.074955054900539e+149

    1. Initial program 18.8

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

    if 2.074955054900539e+149 < 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 8.1

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

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 2.409964474129751 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 5.23044770692996771 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 2.0749550549005391 \cdot 10^{149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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)))))