Average Error: 31.2 → 17.8
Time: 33.4s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.9743900982234307 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log base \cdot \frac{-1}{2}}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le -1.7951643962804085 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log base}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \le 1.9124118988295903 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log base}{\log im} \cdot \frac{1}{2}}\\ \mathbf{elif}\;re \le 5.201817549621194 \cdot 10^{+60}:\\ \;\;\;\;\frac{\log \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \log base}{\log base \cdot \log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

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 5 regimes

  2. if re < -3.9743900982234307e+83

    1. Initial program 46.8

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Initial simplification46.8

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

    4. Applied times-frac46.7

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

    5. Simplified46.7

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

    7. Applied pow1/246.7

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

    8. Applied log-pow46.7

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

    9. Applied associate-/l*46.8

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

    10. Taylor expanded around -inf 62.8

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

    11. Simplified10.2

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

    if -3.9743900982234307e+83 < re < -1.7951643962804085e-238

    1. Initial program 20.1

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Initial simplification20.1

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

    4. Applied times-frac20.0

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

    5. Simplified20.0

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

    7. Applied pow1/220.0

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

    8. Applied log-pow20.0

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

    9. Applied associate-/l*20.0

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

    if -1.7951643962804085e-238 < re < 1.9124118988295903e-230

    1. Initial program 30.8

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Initial simplification30.8

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

    4. Applied times-frac30.7

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

    5. Simplified30.7

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

    7. Applied pow1/230.7

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

    8. Applied log-pow30.7

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

    9. Applied associate-/l*30.8

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

    10. Taylor expanded around 0 32.0

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{1}{2} \cdot \frac{\log base}{\log im}}} \cdot 1\]

    if 1.9124118988295903e-230 < re < 5.201817549621194e+60

    1. Initial program 19.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Initial simplification19.7

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

    4. Applied add-cube-cbrt19.7

      \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)} \cdot \log base}{\log base \cdot \log base}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt19.7

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

    7. Applied cbrt-prod19.7

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

    8. Applied associate-*l*19.7

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

    if 5.201817549621194e+60 < re

    1. Initial program 44.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Initial simplification44.9

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

    4. Applied times-frac44.8

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

    5. Simplified44.8

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

    6. Taylor expanded around inf 11.1

      \[\leadsto \frac{\log \color{blue}{re}}{\log base} \cdot 1\]
  3. Recombined 5 regimes into one program.

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.9743900982234307 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log base \cdot \frac{-1}{2}}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le -1.7951643962804085 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log base}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \le 1.9124118988295903 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{\log base}{\log im} \cdot \frac{1}{2}}\\ \mathbf{elif}\;re \le 5.201817549621194 \cdot 10^{+60}:\\ \;\;\;\;\frac{\log \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} \cdot \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \log base}{\log base \cdot \log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Runtime

Time bar (total: 33.4s)Debug logProfile

herbie shell --seed 2018249 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0))))