Average Error: 31.6 → 17.4
Time: 11.4s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.224721482189337913047315552869611433081 \cdot 10^{153}:\\ \;\;\;\;\log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 264502897229656192:\\ \;\;\;\;\log \left({\left({\left(re \cdot re + im \cdot im\right)}^{\left(\sqrt{\frac{1}{2}}\right)}\right)}^{\left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.224721482189337913047315552869611433081 \cdot 10^{153}:\\
\;\;\;\;\log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\

\mathbf{elif}\;re \le 264502897229656192:\\
\;\;\;\;\log \left({\left({\left(re \cdot re + im \cdot im\right)}^{\left(\sqrt{\frac{1}{2}}\right)}\right)}^{\left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\

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

\end{array}
double f(double re, double im) {
        double r98073 = re;
        double r98074 = r98073 * r98073;
        double r98075 = im;
        double r98076 = r98075 * r98075;
        double r98077 = r98074 + r98076;
        double r98078 = sqrt(r98077);
        double r98079 = log(r98078);
        double r98080 = 10.0;
        double r98081 = log(r98080);
        double r98082 = r98079 / r98081;
        return r98082;
}
↓
double f(double re, double im) {
        double r98083 = re;
        double r98084 = -5.224721482189338e+153;
        bool r98085 = r98083 <= r98084;
        double r98086 = -1.0;
        double r98087 = r98086 / r98083;
        double r98088 = 1.0;
        double r98089 = 10.0;
        double r98090 = log(r98089);
        double r98091 = r98088 / r98090;
        double r98092 = sqrt(r98091);
        double r98093 = -r98092;
        double r98094 = pow(r98087, r98093);
        double r98095 = log(r98094);
        double r98096 = sqrt(r98090);
        double r98097 = r98088 / r98096;
        double r98098 = r98095 * r98097;
        double r98099 = 2.645028972296562e+17;
        bool r98100 = r98083 <= r98099;
        double r98101 = r98083 * r98083;
        double r98102 = im;
        double r98103 = r98102 * r98102;
        double r98104 = r98101 + r98103;
        double r98105 = 0.5;
        double r98106 = sqrt(r98105);
        double r98107 = pow(r98104, r98106);
        double r98108 = r98106 / r98096;
        double r98109 = pow(r98107, r98108);
        double r98110 = log(r98109);
        double r98111 = r98110 * r98097;
        double r98112 = log(r98083);
        double r98113 = r98112 * r98092;
        double r98114 = r98113 * r98097;
        double r98115 = r98100 ? r98111 : r98114;
        double r98116 = r98085 ? r98098 : r98115;
        return r98116;
}

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 < -5.224721482189338e+153

    1. Initial program 64.0

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

    3. Applied add-sqr-sqrt64.0

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

    4. Applied pow1/264.0

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

    5. Applied log-pow64.0

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

    6. Applied times-frac64.0

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

    8. Applied div-inv64.0

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

    9. Applied associate-*r*64.0

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

    11. Applied add-log-exp64.0

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

    12. Simplified64.0

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

    13. Taylor expanded around -inf 6.5

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

    14. Simplified6.5

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

    if -5.224721482189338e+153 < re < 2.645028972296562e+17

    1. Initial program 21.7

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

    3. Applied add-sqr-sqrt21.7

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

    4. Applied pow1/221.7

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

    5. Applied log-pow21.7

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

    6. Applied times-frac21.7

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

    8. Applied div-inv21.5

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

    9. Applied associate-*r*21.5

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

    11. Applied add-log-exp21.6

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

    12. Simplified21.5

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

    14. Applied pow121.5

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

    15. Applied log-pow21.5

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

    16. Applied sqrt-prod21.5

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

    17. Applied add-sqr-sqrt21.9

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

    18. Applied times-frac21.5

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

    19. Applied pow-unpow21.5

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

    20. Simplified21.5

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

    if 2.645028972296562e+17 < re

    1. Initial program 41.4

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

    3. Applied add-sqr-sqrt41.4

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

    4. Applied pow1/241.4

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

    5. Applied log-pow41.4

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

    6. Applied times-frac41.4

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

    8. Applied div-inv41.3

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

    9. Applied associate-*r*41.3

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

    10. Taylor expanded around inf 12.1

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

    11. Simplified12.1

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.224721482189337913047315552869611433081 \cdot 10^{153}:\\ \;\;\;\;\log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 264502897229656192:\\ \;\;\;\;\log \left({\left({\left(re \cdot re + im \cdot im\right)}^{\left(\sqrt{\frac{1}{2}}\right)}\right)}^{\left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\left(\log re \cdot \sqrt{\frac{1}{\log 10}}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))