Average Error: 32.7 → 9.3
Time: 27.0s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -2759744.9040251518599689006805419921875:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -5.32610268211259449984213034556843154695 \cdot 10^{-30}:\\ \;\;\;\;e^{\log \left(\log \left(e^{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -4.399901143579185647092718121671803160377 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]
e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)
↓
\begin{array}{l}
\mathbf{if}\;x.re \le -2759744.9040251518599689006805419921875:\\
\;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{elif}\;x.re \le -5.32610268211259449984213034556843154695 \cdot 10^{-30}:\\
\;\;\;\;e^{\log \left(\log \left(e^{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{elif}\;x.re \le -4.399901143579185647092718121671803160377 \cdot 10^{-310}:\\
\;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{else}:\\
\;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1384235 = x_re;
        double r1384236 = r1384235 * r1384235;
        double r1384237 = x_im;
        double r1384238 = r1384237 * r1384237;
        double r1384239 = r1384236 + r1384238;
        double r1384240 = sqrt(r1384239);
        double r1384241 = log(r1384240);
        double r1384242 = y_re;
        double r1384243 = r1384241 * r1384242;
        double r1384244 = atan2(r1384237, r1384235);
        double r1384245 = y_im;
        double r1384246 = r1384244 * r1384245;
        double r1384247 = r1384243 - r1384246;
        double r1384248 = exp(r1384247);
        double r1384249 = r1384241 * r1384245;
        double r1384250 = r1384244 * r1384242;
        double r1384251 = r1384249 + r1384250;
        double r1384252 = cos(r1384251);
        double r1384253 = r1384248 * r1384252;
        return r1384253;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1384254 = x_re;
        double r1384255 = -2759744.904025152;
        bool r1384256 = r1384254 <= r1384255;
        double r1384257 = -r1384254;
        double r1384258 = log(r1384257);
        double r1384259 = y_re;
        double r1384260 = r1384258 * r1384259;
        double r1384261 = y_im;
        double r1384262 = x_im;
        double r1384263 = atan2(r1384262, r1384254);
        double r1384264 = r1384261 * r1384263;
        double r1384265 = r1384260 - r1384264;
        double r1384266 = exp(r1384265);
        double r1384267 = -5.3261026821125945e-30;
        bool r1384268 = r1384254 <= r1384267;
        double r1384269 = r1384254 * r1384254;
        double r1384270 = r1384262 * r1384262;
        double r1384271 = r1384269 + r1384270;
        double r1384272 = sqrt(r1384271);
        double r1384273 = exp(r1384272);
        double r1384274 = log(r1384273);
        double r1384275 = log(r1384274);
        double r1384276 = r1384275 * r1384259;
        double r1384277 = r1384276 - r1384264;
        double r1384278 = exp(r1384277);
        double r1384279 = -4.3999011435792e-310;
        bool r1384280 = r1384254 <= r1384279;
        double r1384281 = log(r1384254);
        double r1384282 = r1384259 * r1384281;
        double r1384283 = r1384282 - r1384264;
        double r1384284 = exp(r1384283);
        double r1384285 = r1384280 ? r1384266 : r1384284;
        double r1384286 = r1384268 ? r1384278 : r1384285;
        double r1384287 = r1384256 ? r1384266 : r1384286;
        return r1384287;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x.re < -2759744.904025152 or -5.3261026821125945e-30 < x.re < -4.3999011435792e-310

    1. Initial program 32.6

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around 0 18.2

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]

    3. Taylor expanded around -inf 6.4

      \[\leadsto e^{\log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

    4. Simplified6.4

      \[\leadsto e^{\log \color{blue}{\left(-x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

    if -2759744.904025152 < x.re < -5.3261026821125945e-30

    1. Initial program 12.6

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around 0 6.7

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]
    3. Using strategy rm

    4. Applied add-log-exp20.8

      \[\leadsto e^{\log \color{blue}{\left(\log \left(e^{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

    if -4.3999011435792e-310 < x.re

    1. Initial program 33.9

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around 0 20.9

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{1}\]

    3. Taylor expanded around inf 11.4

      \[\leadsto e^{\log \color{blue}{x.re} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -2759744.9040251518599689006805419921875:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -5.32610268211259449984213034556843154695 \cdot 10^{-30}:\\ \;\;\;\;e^{\log \left(\log \left(e^{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -4.399901143579185647092718121671803160377 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))