Average Error: 32.3 → 18.2
Time: 18.0s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.306028001854704 \cdot 10^{91}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -5.86817855579856565 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{\frac{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}}\\ \mathbf{elif}\;re \le 1.0322374213733688 \cdot 10^{-184}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 9.218569920511639 \cdot 10^{129}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{{\left(\log base \cdot \log base\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}} \cdot \left(\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - \left(\log base \cdot \log base\right) \cdot \left(0.0 \cdot 0.0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.306028001854704 \cdot 10^{91}:\\
\;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\

\mathbf{elif}\;re \le -5.86817855579856565 \cdot 10^{-226}:\\
\;\;\;\;\frac{1}{\frac{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}}\\

\mathbf{elif}\;re \le 1.0322374213733688 \cdot 10^{-184}:\\
\;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\

\mathbf{elif}\;re \le 9.218569920511639 \cdot 10^{129}:\\
\;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{{\left(\log base \cdot \log base\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}} \cdot \left(\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - \left(\log base \cdot \log base\right) \cdot \left(0.0 \cdot 0.0\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\end{array}
double f(double re, double im, double base) {
        double r222 = re;
        double r223 = r222 * r222;
        double r224 = im;
        double r225 = r224 * r224;
        double r226 = r223 + r225;
        double r227 = sqrt(r226);
        double r228 = log(r227);
        double r229 = base;
        double r230 = log(r229);
        double r231 = r228 * r230;
        double r232 = atan2(r224, r222);
        double r233 = 0.0;
        double r234 = r232 * r233;
        double r235 = r231 + r234;
        double r236 = r230 * r230;
        double r237 = r233 * r233;
        double r238 = r236 + r237;
        double r239 = r235 / r238;
        return r239;
}
↓
double f(double re, double im, double base) {
        double r240 = re;
        double r241 = -5.306028001854704e+91;
        bool r242 = r240 <= r241;
        double r243 = -1.0;
        double r244 = r243 * r240;
        double r245 = log(r244);
        double r246 = base;
        double r247 = log(r246);
        double r248 = r245 * r247;
        double r249 = im;
        double r250 = atan2(r249, r240);
        double r251 = 0.0;
        double r252 = r250 * r251;
        double r253 = r248 + r252;
        double r254 = r247 * r247;
        double r255 = r251 * r251;
        double r256 = r254 + r255;
        double r257 = r253 / r256;
        double r258 = -5.868178555798566e-226;
        bool r259 = r240 <= r258;
        double r260 = 1.0;
        double r261 = 2.0;
        double r262 = pow(r247, r261);
        double r263 = r255 + r262;
        double r264 = r240 * r240;
        double r265 = r249 * r249;
        double r266 = r264 + r265;
        double r267 = sqrt(r266);
        double r268 = log(r267);
        double r269 = r268 * r247;
        double r270 = r269 + r252;
        double r271 = r263 / r270;
        double r272 = r260 / r271;
        double r273 = 1.0322374213733688e-184;
        bool r274 = r240 <= r273;
        double r275 = log(r249);
        double r276 = r275 * r247;
        double r277 = r276 + r252;
        double r278 = r277 / r256;
        double r279 = 9.218569920511639e+129;
        bool r280 = r240 <= r279;
        double r281 = 3.0;
        double r282 = pow(r254, r281);
        double r283 = pow(r255, r281);
        double r284 = r282 + r283;
        double r285 = r270 / r284;
        double r286 = r254 * r254;
        double r287 = r255 * r255;
        double r288 = r254 * r255;
        double r289 = r287 - r288;
        double r290 = r286 + r289;
        double r291 = r285 * r290;
        double r292 = sqrt(r256);
        double r293 = r260 / r292;
        double r294 = log(r240);
        double r295 = r294 * r247;
        double r296 = r295 + r252;
        double r297 = r296 / r292;
        double r298 = r293 * r297;
        double r299 = r280 ? r291 : r298;
        double r300 = r274 ? r278 : r299;
        double r301 = r259 ? r272 : r300;
        double r302 = r242 ? r257 : r301;
        return r302;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if re < -5.306028001854704e+91

    1. Initial program 50.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    2. Taylor expanded around -inf 8.3

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    if -5.306028001854704e+91 < re < -5.868178555798566e-226

    1. Initial program 20.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
    2. Using strategy rm

    3. Applied clear-num20.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\log base \cdot \log base + 0.0 \cdot 0.0}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}}}\]

    4. Simplified20.1

      \[\leadsto \frac{1}{\color{blue}{\frac{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}}}\]

    if -5.868178555798566e-226 < re < 1.0322374213733688e-184

    1. Initial program 31.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    2. Taylor expanded around 0 34.9

      \[\leadsto \frac{\log \color{blue}{im} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    if 1.0322374213733688e-184 < re < 9.218569920511639e+129

    1. Initial program 17.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
    2. Using strategy rm

    3. Applied flip3-+18.0

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\frac{{\left(\log base \cdot \log base\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}}{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - \left(\log base \cdot \log base\right) \cdot \left(0.0 \cdot 0.0\right)\right)}}}\]

    4. Applied associate-/r/18.0

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{{\left(\log base \cdot \log base\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}} \cdot \left(\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - \left(\log base \cdot \log base\right) \cdot \left(0.0 \cdot 0.0\right)\right)\right)}\]

    if 9.218569920511639e+129 < re

    1. Initial program 57.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt57.5

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

    4. Applied *-un-lft-identity57.5

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    5. Applied times-frac57.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

    6. Taylor expanded around inf 7.0

      \[\leadsto \frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \color{blue}{re} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.306028001854704 \cdot 10^{91}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -5.86817855579856565 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{\frac{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}}\\ \mathbf{elif}\;re \le 1.0322374213733688 \cdot 10^{-184}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 9.218569920511639 \cdot 10^{129}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{{\left(\log base \cdot \log base\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}} \cdot \left(\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - \left(\log base \cdot \log base\right) \cdot \left(0.0 \cdot 0.0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))