Average Error: 30.5 → 16.8
Time: 3.3m
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-1}{im} \le -3.009688101439238 \cdot 10^{-87}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\\ \mathbf{if}\;\frac{-1}{im} \le -3.775683847004391 \cdot 10^{-307}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{if}\;\frac{-1}{im} \le 7.29346016288141 \cdot 10^{-105}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log base \cdot \log base}}\\ \end{array}\]

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 (/ -1 im) < -3.009688101439238e-87

    1. Initial program 20.3

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

    if -3.009688101439238e-87 < (/ -1 im) < -3.775683847004391e-307

    1. Initial program 47.5

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

    2. Taylor expanded around 0 9.6

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

    if -3.775683847004391e-307 < (/ -1 im) < 7.29346016288141e-105

    1. Initial program 50.3

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

    3. Applied add-cbrt-cube50.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \color{blue}{\sqrt[3]{\left(\log base \cdot \log base\right) \cdot \log base}} + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    4. Applied add-cbrt-cube50.4

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}} \cdot \sqrt[3]{\left(\log base \cdot \log base\right) \cdot \log base} + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    5. Applied cbrt-unprod50.4

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(\left(\log base \cdot \log base\right) \cdot \log base\right)}} + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    6. Applied simplify50.4

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^{3}}} + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    7. Taylor expanded around -inf 9.4

      \[\leadsto \frac{\sqrt[3]{{\left(\log base \cdot \log \color{blue}{\left(-1 \cdot im\right)}\right)}^{3}} + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    8. Applied simplify9.1

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

    if 7.29346016288141e-105 < (/ -1 im)

    1. Initial program 21.1

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

    3. Applied add-sqr-sqrt21.1

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

    4. Applied *-un-lft-identity21.1

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

    5. Applied times-frac21.1

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

    6. Applied simplify21.1

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

    7. Applied simplify21.1

      \[\leadsto \frac{1}{\sqrt{\log base \cdot \log base}} \cdot \color{blue}{\frac{\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\sqrt{\log base \cdot \log base}}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify16.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{-1}{im} \le -3.009688101439238 \cdot 10^{-87}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\\ \mathbf{if}\;\frac{-1}{im} \le -3.775683847004391 \cdot 10^{-307}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{if}\;\frac{-1}{im} \le 7.29346016288141 \cdot 10^{-105}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base}} \cdot \frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log base \cdot \log base}}\\ \end{array}}\]

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' 
(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))))