Average Error: 30.9 → 17.3
Time: 1.1m
Precision: 64
Internal Precision: 320
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -2.8281587110155982 \cdot 10^{+128}:\\ \;\;\;\;\frac{-\log base}{\frac{\log base \cdot \log base}{\log \left(\frac{-1}{im}\right)}}\\ \mathbf{if}\;im \le -7.55394099205308 \cdot 10^{-303}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\sqrt{\log base \cdot \log base}}\\ \mathbf{if}\;im \le 8.149399807342678 \cdot 10^{-186}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{if}\;im \le 1.202536316015387 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\sqrt{\log base \cdot \log base}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

  1. Inputs

  2. Original Output:

    Herbie Output:

Derivation

  1. Split input into 4 regimes

  2. if im < -2.8281587110155982e+128

    1. Initial program 54.6

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

    3. Applied add-sqr-sqrt54.6

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

    4. Applied *-un-lft-identity54.6

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

    5. Applied times-frac54.6

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

    6. Applied simplify54.6

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

    7. Applied simplify54.6

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

    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 simplify7.8

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

    if -2.8281587110155982e+128 < im < -7.55394099205308e-303 or 8.149399807342678e-186 < im < 1.202536316015387e+124

    1. Initial program 19.3

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

    3. Applied add-sqr-sqrt19.3

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

    4. Applied *-un-lft-identity19.3

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

    5. Applied times-frac19.3

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

    6. Applied simplify19.3

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

    7. Applied simplify19.3

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

    if -7.55394099205308e-303 < im < 8.149399807342678e-186

    1. Initial program 30.2

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

    2. Taylor expanded around -inf 33.5

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

    3. Applied simplify33.4

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

    if 1.202536316015387e+124 < im

    1. Initial program 54.7

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

    2. Taylor expanded around 0 7.6

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' 
(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))))