Average Error: 31.5 → 18.1
Time: 1.2m
Precision: 64
Internal Precision: 384
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-1}{im} \le -1.5190513189386892 \cdot 10^{+286}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{if}\;\frac{-1}{im} \le -2.985273544946111 \cdot 10^{+190}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \mathbf{if}\;\frac{-1}{im} \le -2.0300191434495103 \cdot 10^{-149}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\sqrt{\log base \cdot \log base}}\\ \mathbf{if}\;\frac{-1}{im} \le 3.611722326507118 \cdot 10^{-307}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{if}\;\frac{-1}{im} \le 4.847019719982004 \cdot 10^{-28}:\\ \;\;\;\;\frac{\left(-\log base\right) \cdot \log \left(\frac{-1}{im}\right)}{\left|\log base\right| \cdot \left|\log base\right|}\\ \mathbf{if}\;\frac{-1}{im} \le 9.017275512246256 \cdot 10^{+201}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\sqrt{\log base \cdot \log base}}\\ \mathbf{if}\;\frac{-1}{im} \le 6.571247444903276 \cdot 10^{+269}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\sqrt{\log base \cdot \log base}}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Split input into 5 regimes

  2. if (/ -1 im) < -1.5190513189386892e+286 or 9.017275512246256e+201 < (/ -1 im) < 6.571247444903276e+269

    1. Initial program 32.5

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

    2. Taylor expanded around -inf 34.4

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

    3. Applied simplify34.3

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

    if -1.5190513189386892e+286 < (/ -1 im) < -2.985273544946111e+190

    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. Taylor expanded around inf 34.5

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

    3. Applied simplify34.5

      \[\leadsto \color{blue}{\frac{-\log re}{-\log base}}\]

    if -2.985273544946111e+190 < (/ -1 im) < -2.0300191434495103e-149 or 4.847019719982004e-28 < (/ -1 im) < 9.017275512246256e+201 or 6.571247444903276e+269 < (/ -1 im)

    1. Initial program 18.6

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

    3. Applied add-sqr-sqrt18.6

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

    4. Applied *-un-lft-identity18.6

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

    5. Applied times-frac18.6

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

    6. Applied simplify18.6

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

    7. Applied simplify18.6

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

    if -2.0300191434495103e-149 < (/ -1 im) < 3.611722326507118e-307

    1. Initial program 60.8

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

    2. Taylor expanded around 0 7.0

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

    if 3.611722326507118e-307 < (/ -1 im) < 4.847019719982004e-28

    1. Initial program 41.3

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

    3. Applied add-sqr-sqrt41.3

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

    4. Applied *-un-lft-identity41.3

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

    5. Applied times-frac41.3

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

    6. Applied simplify41.3

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

    7. Applied simplify41.3

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

    8. Taylor expanded around -inf 62.8

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

    9. Applied simplify12.7

      \[\leadsto \color{blue}{\frac{\left(-\log base\right) \cdot \log \left(\frac{-1}{im}\right)}{\left|\log base\right| \cdot \left|\log base\right|}}\]
  3. Recombined 5 regimes into one program.

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' 
(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))))