Average Error: 31.1 → 17.8
Time: 35.4s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.0260633371563236 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\left({\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \log \left(\frac{-1}{re}\right)\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le -1.6983590970193986 \cdot 10^{-164}:\\ \;\;\;\;\frac{\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -3.151634490341881 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\left({\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \log \left(\frac{-1}{re}\right)\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le 1.1887108504340247 \cdot 10^{+149}:\\ \;\;\;\;\frac{\log \left(im \cdot im + re \cdot re\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\left({\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}} \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot -2\right)\\ \end{array}\]

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 < -3.0260633371563236e+95 or -1.6983590970193986e-164 < re < -3.151634490341881e-222

    1. Initial program 45.5

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

    3. Applied add-sqr-sqrt45.5

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

    4. Applied pow1/245.5

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

    5. Applied log-pow45.5

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

    6. Applied times-frac45.4

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

    8. Applied add-sqr-sqrt45.4

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

    9. Applied associate-*l*45.4

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

    10. Taylor expanded around -inf 17.4

      \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \color{blue}{\left(-2 \cdot \left(\left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}}\right)\right)}\]

    if -3.0260633371563236e+95 < re < -1.6983590970193986e-164

    1. Initial program 15.9

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

    3. Applied add-sqr-sqrt15.9

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

    4. Applied pow1/215.9

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

    5. Applied log-pow15.9

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

    6. Applied times-frac15.9

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

    8. Applied associate-*r/15.9

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

    if -3.151634490341881e-222 < re < 1.1887108504340247e+149

    1. Initial program 22.4

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

    3. Applied add-sqr-sqrt22.4

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

    4. Applied pow1/222.4

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

    5. Applied log-pow22.4

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

    6. Applied times-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 add-sqr-sqrt22.4

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

    9. Applied associate-*l*22.3

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

    11. Applied associate-*r/22.3

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

    if 1.1887108504340247e+149 < re

    1. Initial program 60.5

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

    3. Applied add-sqr-sqrt60.5

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

    4. Applied pow1/260.5

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

    5. Applied log-pow60.5

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

    6. Applied times-frac60.5

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

    8. Applied add-sqr-sqrt60.5

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

    9. Applied associate-*l*60.5

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

    10. Taylor expanded around inf 6.8

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.0260633371563236 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\left({\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \log \left(\frac{-1}{re}\right)\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le -1.6983590970193986 \cdot 10^{-164}:\\ \;\;\;\;\frac{\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -3.151634490341881 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\left({\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \log \left(\frac{-1}{re}\right)\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le 1.1887108504340247 \cdot 10^{+149}:\\ \;\;\;\;\frac{\log \left(im \cdot im + re \cdot re\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\left({\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}} \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot -2\right)\\ \end{array}\]

Runtime

Time bar (total: 35.4s)Debug logProfile

herbie shell --seed 2018249 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))