Average Error: 30.3 → 16.6
Time: 1.7m
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.2068211949372295 \cdot 10^{+73}:\\ \;\;\;\;\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}{\log base}}{\log base}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if re < -1.2068211949372295e+73

    1. Initial program 45.6

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

    2. Initial simplification45.6

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

    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. Simplified9.8

      \[\leadsto \color{blue}{\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)}\]
    5. Using strategy rm

    6. Applied add-exp-log9.8

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

    7. Applied rem-log-exp9.8

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

    if -1.2068211949372295e+73 < re

    1. Initial program 20.9

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

    2. Initial simplification20.9

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

    4. Applied associate-/r*20.9

      \[\leadsto \color{blue}{\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base}}{\log base}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2068211949372295 \cdot 10^{+73}:\\ \;\;\;\;\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}{\log base}}{\log base}\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018348 
(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))))