Average Error: 31.9 → 17.8
Time: 19.4s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.366950917668637798933900247544484583658 \cdot 10^{69}:\\ \;\;\;\;\left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\ \;\;\;\;\left(\sqrt{\frac{1}{2}} \cdot \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.366950917668637798933900247544484583658 \cdot 10^{69}:\\
\;\;\;\;\left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\

\mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\
\;\;\;\;\left(\sqrt{\frac{1}{2}} \cdot \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\

\end{array}
double f(double re, double im) {
        double r1408943 = re;
        double r1408944 = r1408943 * r1408943;
        double r1408945 = im;
        double r1408946 = r1408945 * r1408945;
        double r1408947 = r1408944 + r1408946;
        double r1408948 = sqrt(r1408947);
        double r1408949 = log(r1408948);
        double r1408950 = 10.0;
        double r1408951 = log(r1408950);
        double r1408952 = r1408949 / r1408951;
        return r1408952;
}
↓
double f(double re, double im) {
        double r1408953 = re;
        double r1408954 = -2.3669509176686378e+69;
        bool r1408955 = r1408953 <= r1408954;
        double r1408956 = -1.0;
        double r1408957 = r1408956 / r1408953;
        double r1408958 = log(r1408957);
        double r1408959 = -2.0;
        double r1408960 = r1408958 * r1408959;
        double r1408961 = 0.5;
        double r1408962 = 10.0;
        double r1408963 = log(r1408962);
        double r1408964 = sqrt(r1408963);
        double r1408965 = r1408961 / r1408964;
        double r1408966 = r1408960 * r1408965;
        double r1408967 = 1.0;
        double r1408968 = r1408967 / r1408964;
        double r1408969 = r1408966 * r1408968;
        double r1408970 = 7.747777771049568e+94;
        bool r1408971 = r1408953 <= r1408970;
        double r1408972 = sqrt(r1408961);
        double r1408973 = r1408972 / r1408964;
        double r1408974 = im;
        double r1408975 = r1408974 * r1408974;
        double r1408976 = r1408953 * r1408953;
        double r1408977 = r1408975 + r1408976;
        double r1408978 = log(r1408977);
        double r1408979 = r1408973 * r1408978;
        double r1408980 = r1408972 * r1408979;
        double r1408981 = r1408980 * r1408968;
        double r1408982 = r1408967 / r1408963;
        double r1408983 = sqrt(r1408982);
        double r1408984 = log(r1408953);
        double r1408985 = r1408983 * r1408984;
        double r1408986 = r1408985 * r1408968;
        double r1408987 = r1408971 ? r1408981 : r1408986;
        double r1408988 = r1408955 ? r1408969 : r1408987;
        return r1408988;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -2.3669509176686378e+69

    1. Initial program 46.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt46.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

    4. Applied pow1/246.4

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow46.4

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac46.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
    7. Using strategy rm

    8. Applied div-inv46.4

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]

    9. Applied associate-*r*46.4

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}}\]

    10. Taylor expanded around -inf 10.6

      \[\leadsto \left(\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\]

    11. Simplified10.6

      \[\leadsto \left(\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\frac{-1}{re}\right) \cdot -2\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\]

    if -2.3669509176686378e+69 < re < 7.747777771049568e+94

    1. Initial program 22.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt22.5

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

    4. Applied pow1/222.5

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow22.5

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac22.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
    7. Using strategy rm

    8. Applied div-inv22.3

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]

    9. Applied associate-*r*22.3

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}}\]
    10. Using strategy rm

    11. Applied *-un-lft-identity22.3

      \[\leadsto \left(\frac{\frac{1}{2}}{\color{blue}{1 \cdot \sqrt{\log 10}}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\]

    12. Applied add-sqr-sqrt22.6

      \[\leadsto \left(\frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{1 \cdot \sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\]

    13. Applied times-frac22.3

      \[\leadsto \left(\color{blue}{\left(\frac{\sqrt{\frac{1}{2}}}{1} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\]

    14. Applied associate-*l*22.4

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{1}{2}}}{1} \cdot \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\right)} \cdot \frac{1}{\sqrt{\log 10}}\]

    if 7.747777771049568e+94 < re

    1. Initial program 49.1

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt49.1

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

    4. Applied pow1/249.1

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow49.1

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac49.1

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
    7. Using strategy rm

    8. Applied div-inv49.1

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]

    9. Applied associate-*r*49.1

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}}\]

    10. Taylor expanded around inf 9.9

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{1}{re}\right)\right)\right)} \cdot \frac{1}{\sqrt{\log 10}}\]

    11. Simplified9.9

      \[\leadsto \color{blue}{\left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right)} \cdot \frac{1}{\sqrt{\log 10}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.366950917668637798933900247544484583658 \cdot 10^{69}:\\ \;\;\;\;\left(\left(\log \left(\frac{-1}{re}\right) \cdot -2\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\ \;\;\;\;\left(\sqrt{\frac{1}{2}} \cdot \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))