Average Error: 38.1 → 19.9
Time: 4.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.8408794472488761 \cdot 10^{164}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-2 \cdot re}}\\ \mathbf{elif}\;re \le 9.76899627496736849 \cdot 10^{-287}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\\ \mathbf{elif}\;re \le 1.9399661775247916 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.8408794472488761 \cdot 10^{164}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-2 \cdot re}}\\

\mathbf{elif}\;re \le 9.76899627496736849 \cdot 10^{-287}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\\

\mathbf{elif}\;re \le 1.9399661775247916 \cdot 10^{118}:\\
\;\;\;\;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 r295206 = 0.5;
        double r295207 = 2.0;
        double r295208 = re;
        double r295209 = r295208 * r295208;
        double r295210 = im;
        double r295211 = r295210 * r295210;
        double r295212 = r295209 + r295211;
        double r295213 = sqrt(r295212);
        double r295214 = r295213 + r295208;
        double r295215 = r295207 * r295214;
        double r295216 = sqrt(r295215);
        double r295217 = r295206 * r295216;
        return r295217;
}
↓
double f(double re, double im) {
        double r295218 = re;
        double r295219 = -5.840879447248876e+164;
        bool r295220 = r295218 <= r295219;
        double r295221 = 0.5;
        double r295222 = 2.0;
        double r295223 = im;
        double r295224 = r295223 * r295223;
        double r295225 = -2.0;
        double r295226 = r295225 * r295218;
        double r295227 = r295224 / r295226;
        double r295228 = r295222 * r295227;
        double r295229 = sqrt(r295228);
        double r295230 = r295221 * r295229;
        double r295231 = 9.768996274967368e-287;
        bool r295232 = r295218 <= r295231;
        double r295233 = sqrt(r295222);
        double r295234 = r295218 * r295218;
        double r295235 = r295234 + r295224;
        double r295236 = sqrt(r295235);
        double r295237 = r295236 - r295218;
        double r295238 = sqrt(r295237);
        double r295239 = r295223 / r295238;
        double r295240 = fabs(r295239);
        double r295241 = r295233 * r295240;
        double r295242 = r295221 * r295241;
        double r295243 = 1.9399661775247916e+118;
        bool r295244 = r295218 <= r295243;
        double r295245 = r295236 + r295218;
        double r295246 = r295222 * r295245;
        double r295247 = sqrt(r295246);
        double r295248 = r295221 * r295247;
        double r295249 = 2.0;
        double r295250 = r295249 * r295218;
        double r295251 = r295222 * r295250;
        double r295252 = sqrt(r295251);
        double r295253 = r295221 * r295252;
        double r295254 = r295244 ? r295248 : r295253;
        double r295255 = r295232 ? r295242 : r295254;
        double r295256 = r295220 ? r295230 : r295255;
        return r295256;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.1
Target33.2
Herbie19.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -5.840879447248876e+164

    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. Simplified50.7

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

    5. Taylor expanded around -inf 30.3

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

    if -5.840879447248876e+164 < re < 9.768996274967368e-287

    1. Initial program 39.1

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

    3. Applied flip-+39.0

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

    4. Simplified30.7

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

    6. Applied add-sqr-sqrt30.8

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

    7. Applied times-frac28.8

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

    9. Applied sqrt-prod28.9

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

    10. Simplified20.8

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

    if 9.768996274967368e-287 < re < 1.9399661775247916e+118

    1. Initial program 19.7

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

    if 1.9399661775247916e+118 < re

    1. Initial program 56.0

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

    2. Taylor expanded around inf 10.2

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

  4. Final simplification19.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.8408794472488761 \cdot 10^{164}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-2 \cdot re}}\\ \mathbf{elif}\;re \le 9.76899627496736849 \cdot 10^{-287}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\\ \mathbf{elif}\;re \le 1.9399661775247916 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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)))))