Average Error: 31.1 → 17.4
Time: 39.3s
Precision: 64
Internal Precision: 320
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.4847177066776937 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le -7.077682315366247 \cdot 10^{-246}:\\ \;\;\;\;\frac{\log \left(\sqrt{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \left|\sqrt[3]{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right|\right) \cdot \log base}{\log base \cdot \log base}\\ \mathbf{elif}\;re \le 3.5882029693490434 \cdot 10^{-272}:\\ \;\;\;\;\frac{\log base \cdot \log im}{\log base \cdot \log base}\\ \mathbf{elif}\;re \le 1.997968920449443 \cdot 10^{+69}:\\ \;\;\;\;\frac{\log \left(\sqrt{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \left|\sqrt[3]{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right|\right) \cdot \log base}{\log base \cdot \log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

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 4 regimes

  2. if re < -2.4847177066776937e+60

    1. Initial program 44.2

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

    2. Initial simplification44.2

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

    3. Taylor expanded around -inf 62.8

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

    4. Simplified11.1

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

    if -2.4847177066776937e+60 < re < -7.077682315366247e-246 or 3.5882029693490434e-272 < re < 1.997968920449443e+69

    1. Initial program 20.3

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

    2. Initial simplification20.3

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

    4. Applied add-cube-cbrt20.3

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

    5. Applied sqrt-prod20.3

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

    6. Simplified20.3

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

    8. Applied add-sqr-sqrt20.3

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

    9. Applied cbrt-prod20.3

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

    if -7.077682315366247e-246 < re < 3.5882029693490434e-272

    1. Initial program 31.7

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

    2. Initial simplification31.7

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

    3. Taylor expanded around 0 30.3

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

    if 1.997968920449443e+69 < re

    1. Initial program 46.7

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

    2. Initial simplification46.7

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

    3. Taylor expanded around inf 10.6

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

    4. Simplified10.6

      \[\leadsto \color{blue}{\frac{-\log re}{-\log base}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.4847177066776937 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le -7.077682315366247 \cdot 10^{-246}:\\ \;\;\;\;\frac{\log \left(\sqrt{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \left|\sqrt[3]{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right|\right) \cdot \log base}{\log base \cdot \log base}\\ \mathbf{elif}\;re \le 3.5882029693490434 \cdot 10^{-272}:\\ \;\;\;\;\frac{\log base \cdot \log im}{\log base \cdot \log base}\\ \mathbf{elif}\;re \le 1.997968920449443 \cdot 10^{+69}:\\ \;\;\;\;\frac{\log \left(\sqrt{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \left|\sqrt[3]{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right|\right) \cdot \log base}{\log base \cdot \log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

Runtime

Time bar (total: 39.3s)Debug logProfile

herbie shell --seed 2018254 
(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))))