Average Error: 31.2 → 17.4
Time: 4.5m
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.323762613115913 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right)\right)\\ \mathbf{elif}\;re \le -2.531084135726317 \cdot 10^{-272}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{elif}\;re \le 1.861610918855101 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left({\left(\log 10\right)}^{\left(-\frac{1}{2}\right)} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)\right)\\ \mathbf{elif}\;re \le 4.071733907346115 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \end{array}\]

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 4 regimes

  2. if re < -4.323762613115913e+124

    1. Initial program 55.0

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

    3. Applied add-sqr-sqrt55.0

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

    4. Applied pow1/255.0

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

    5. Applied log-pow55.0

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

    6. Applied times-frac54.9

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

    7. Taylor expanded around -inf 7.5

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

    if -4.323762613115913e+124 < re < -2.531084135726317e-272 or 1.861610918855101e-183 < re < 4.071733907346115e+95

    1. Initial program 19.1

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

    3. Applied clear-num19.1

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

    if -2.531084135726317e-272 < re < 1.861610918855101e-183

    1. Initial program 31.4

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

    3. Applied add-sqr-sqrt31.4

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

    4. Applied pow1/231.4

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

    5. Applied log-pow31.4

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

    6. Applied times-frac31.4

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

    8. Applied add-cbrt-cube31.4

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\color{blue}{\sqrt[3]{\left(\sqrt{\log 10} \cdot \sqrt{\log 10}\right) \cdot \sqrt{\log 10}}}}\]

    9. Applied add-cbrt-cube31.4

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\color{blue}{\sqrt[3]{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \log \left(re \cdot re + im \cdot im\right)}}}{\sqrt[3]{\left(\sqrt{\log 10} \cdot \sqrt{\log 10}\right) \cdot \sqrt{\log 10}}}\]

    10. Applied cbrt-undiv31.4

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\sqrt[3]{\frac{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \log \left(re \cdot re + im \cdot im\right)}{\left(\sqrt{\log 10} \cdot \sqrt{\log 10}\right) \cdot \sqrt{\log 10}}}}\]

    11. Simplified31.4

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \sqrt[3]{\color{blue}{{\left(\frac{\log \left(im \cdot im + re \cdot re\right)}{\sqrt{\log 10}}\right)}^{3}}}\]

    12. Taylor expanded around -inf 48.8

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{{\left({\left(-2 \cdot \left(\log \left(\frac{-1}{im}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}^{3}\right)}^{\frac{1}{3}}}\]

    13. Simplified33.3

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

    if 4.071733907346115e+95 < re

    1. Initial program 48.8

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

    3. Applied add-sqr-sqrt48.8

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

    4. Applied pow1/248.8

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

    5. Applied log-pow48.8

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

    6. Applied times-frac48.8

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

    7. Taylor expanded around inf 8.9

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(-2 \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.323762613115913 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right)\right)\\ \mathbf{elif}\;re \le -2.531084135726317 \cdot 10^{-272}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{elif}\;re \le 1.861610918855101 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left({\left(\log 10\right)}^{\left(-\frac{1}{2}\right)} \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)\right)\\ \mathbf{elif}\;re \le 4.071733907346115 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \end{array}\]

Runtime

Time bar (total: 4.5m)Debug logProfile

herbie shell --seed 2018214 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))