Average Error: 31.5 → 8.6
Time: 1.9m
Precision: 64
Internal Precision: 384
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -3.196015042415083 \cdot 10^{+48}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \mathbf{if}\;im \le -4.909241579409979 \cdot 10^{-232}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{im \cdot im + re \cdot re}}\right)}{\log base \cdot \log base}\\ \mathbf{if}\;im \le 4.3518238713509867 \cdot 10^{-156}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{if}\;im \le 4.7136711072338947 \cdot 10^{+120}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{im \cdot im + re \cdot re}}\right)}{\log base \cdot \log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Split input into 4 regimes

  2. if im < -3.196015042415083e+48

    1. Initial program 45.4

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

    2. Applied simplify45.4

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

    3. Taylor expanded around -inf 0.5

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

    4. Applied simplify0.4

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

    if -3.196015042415083e+48 < im < -4.909241579409979e-232 or 4.3518238713509867e-156 < im < 4.7136711072338947e+120

    1. Initial program 17.2

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

    2. Applied simplify17.2

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

    4. Applied add-cube-cbrt17.2

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

    5. Applied sqrt-prod17.2

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

    6. Applied simplify17.2

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

    if -4.909241579409979e-232 < im < 4.3518238713509867e-156

    1. Initial program 30.6

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

    2. Applied simplify30.6

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

    4. Applied add-cbrt-cube30.7

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

    5. Applied simplify30.7

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

    6. Taylor expanded around -inf 0.7

      \[\leadsto \frac{\sqrt[3]{{\left(\log base \cdot \log \color{blue}{\left(-1 \cdot re\right)}\right)}^{3}}}{\log base \cdot \log base}\]

    7. Applied simplify0.4

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

    if 4.7136711072338947e+120 < im

    1. Initial program 53.7

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

    2. Applied simplify53.7

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

    3. Taylor expanded around inf 0.5

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

    4. Applied simplify0.3

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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))))