Average Error: 31.7 → 17.6
Time: 7.9s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.582598474548579396173713851965652835843 \cdot 10^{112}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -1.097857316006781085020055659405458069718 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 5.301603919361751420356837816014607581994 \cdot 10^{-202}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 3.028881099794190950594541308531268747948 \cdot 10^{96}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]
\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 -1.582598474548579396173713851965652835843 \cdot 10^{112}:\\
\;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\mathbf{elif}\;re \le -1.097857316006781085020055659405458069718 \cdot 10^{-283}:\\
\;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\mathbf{elif}\;re \le 5.301603919361751420356837816014607581994 \cdot 10^{-202}:\\
\;\;\;\;\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 3.028881099794190950594541308531268747948 \cdot 10^{96}:\\
\;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\

\end{array}
double f(double re, double im, double base) {
        double r43942 = re;
        double r43943 = r43942 * r43942;
        double r43944 = im;
        double r43945 = r43944 * r43944;
        double r43946 = r43943 + r43945;
        double r43947 = sqrt(r43946);
        double r43948 = log(r43947);
        double r43949 = base;
        double r43950 = log(r43949);
        double r43951 = r43948 * r43950;
        double r43952 = atan2(r43944, r43942);
        double r43953 = 0.0;
        double r43954 = r43952 * r43953;
        double r43955 = r43951 + r43954;
        double r43956 = r43950 * r43950;
        double r43957 = r43953 * r43953;
        double r43958 = r43956 + r43957;
        double r43959 = r43955 / r43958;
        return r43959;
}
↓
double f(double re, double im, double base) {
        double r43960 = re;
        double r43961 = -1.5825984745485794e+112;
        bool r43962 = r43960 <= r43961;
        double r43963 = -1.0;
        double r43964 = r43963 * r43960;
        double r43965 = log(r43964);
        double r43966 = base;
        double r43967 = log(r43966);
        double r43968 = r43965 * r43967;
        double r43969 = im;
        double r43970 = atan2(r43969, r43960);
        double r43971 = 0.0;
        double r43972 = r43970 * r43971;
        double r43973 = r43968 + r43972;
        double r43974 = 2.0;
        double r43975 = cbrt(r43966);
        double r43976 = log(r43975);
        double r43977 = r43974 * r43976;
        double r43978 = r43967 * r43977;
        double r43979 = r43967 * r43976;
        double r43980 = r43978 + r43979;
        double r43981 = r43971 * r43971;
        double r43982 = r43980 + r43981;
        double r43983 = sqrt(r43982);
        double r43984 = r43973 / r43983;
        double r43985 = r43967 * r43967;
        double r43986 = r43985 + r43981;
        double r43987 = sqrt(r43986);
        double r43988 = r43984 / r43987;
        double r43989 = -1.097857316006781e-283;
        bool r43990 = r43960 <= r43989;
        double r43991 = r43960 * r43960;
        double r43992 = r43969 * r43969;
        double r43993 = r43991 + r43992;
        double r43994 = sqrt(r43993);
        double r43995 = log(r43994);
        double r43996 = r43995 * r43967;
        double r43997 = r43996 + r43972;
        double r43998 = r43997 / r43983;
        double r43999 = r43998 / r43987;
        double r44000 = 5.301603919361751e-202;
        bool r44001 = r43960 <= r44000;
        double r44002 = log(r43969);
        double r44003 = r44002 * r43967;
        double r44004 = r44003 + r43972;
        double r44005 = r44004 / r43986;
        double r44006 = 3.028881099794191e+96;
        bool r44007 = r43960 <= r44006;
        double r44008 = 1.0;
        double r44009 = r44008 / r43960;
        double r44010 = log(r44009);
        double r44011 = r44008 / r43966;
        double r44012 = log(r44011);
        double r44013 = r44010 / r44012;
        double r44014 = r44007 ? r43999 : r44013;
        double r44015 = r44001 ? r44005 : r44014;
        double r44016 = r43990 ? r43999 : r44015;
        double r44017 = r43962 ? r43988 : r44016;
        return r44017;
}

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 4 regimes

  2. if re < -1.5825984745485794e+112

    1. Initial program 53.4

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

    3. Applied add-sqr-sqrt53.4

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

    4. Applied associate-/r*53.4

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

    6. Applied add-cube-cbrt53.4

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

    7. Applied log-prod53.4

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

    8. Applied distribute-lft-in53.4

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

    9. Simplified53.4

      \[\leadsto \frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\color{blue}{\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right)} + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    10. Taylor expanded around -inf 8.8

      \[\leadsto \frac{\frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    if -1.5825984745485794e+112 < re < -1.097857316006781e-283 or 5.301603919361751e-202 < re < 3.028881099794191e+96

    1. Initial program 19.7

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

    3. Applied add-sqr-sqrt19.7

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

    4. Applied associate-/r*19.6

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

    6. Applied add-cube-cbrt19.6

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

    7. Applied log-prod19.7

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

    8. Applied distribute-lft-in19.6

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

    9. Simplified19.6

      \[\leadsto \frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\color{blue}{\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right)} + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    if -1.097857316006781e-283 < re < 5.301603919361751e-202

    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 32.2

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

    if 3.028881099794191e+96 < re

    1. Initial program 51.3

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

    2. Taylor expanded around inf 9.6

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.582598474548579396173713851965652835843 \cdot 10^{112}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -1.097857316006781085020055659405458069718 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 5.301603919361751420356837816014607581994 \cdot 10^{-202}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 3.028881099794190950594541308531268747948 \cdot 10^{96}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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))))