Average Error: 32.0 → 17.6
Time: 7.0s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.46186942740825347 \cdot 10^{91}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \log \left(\sqrt[3]{base}\right) + 2 \cdot \left(\log base \cdot \log \left(\sqrt[3]{base}\right)\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -3.6142519675909714 \cdot 10^{-198}:\\ \;\;\;\;\frac{\left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(0.0 \cdot 0.0 + {\left(\log base\right)}^{2}\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}\\ \mathbf{elif}\;re \le 1.09305946911323656 \cdot 10^{-271}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 2.14368816692774462 \cdot 10^{121}:\\ \;\;\;\;\frac{\left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(0.0 \cdot 0.0 + {\left(\log base\right)}^{2}\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Split input into 4 regimes

  2. if re < -3.46186942740825347e91

    1. Initial program 51.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt51.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log \color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)} + 0.0 \cdot 0.0}\]

    4. Applied log-prod51.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \color{blue}{\left(\log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

    5. Applied distribute-lft-in51.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\left(\log base \cdot \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

    6. Simplified51.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\color{blue}{2 \cdot \left(\log base \cdot \log \left(\sqrt[3]{base}\right)\right)} + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

    7. Taylor expanded around -inf 9.6

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(2 \cdot \left(\log base \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

    8. Simplified9.6

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(2 \cdot \left(\log base \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

    if -3.46186942740825347e91 < re < -3.6142519675909714e-198 or 1.09305946911323656e-271 < re < 2.14368816692774462e121

    1. Initial program 19.6

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
    2. Using strategy rm

    3. Applied flip-+19.3

      \[\leadsto \frac{\color{blue}{\frac{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - \tan^{-1}_* \frac{im}{re} \cdot 0.0}}}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    4. Applied associate-/l/19.3

      \[\leadsto \color{blue}{\frac{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(\log base \cdot \log base + 0.0 \cdot 0.0\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}\]

    5. Simplified19.3

      \[\leadsto \frac{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\color{blue}{\left({\left(\log base\right)}^{2} + 0.0 \cdot 0.0\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}\]

    if -3.6142519675909714e-198 < re < 1.09305946911323656e-271

    1. Initial program 31.8

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    2. Taylor expanded around 0 32.8

      \[\leadsto \color{blue}{\frac{\log 1 + \log im}{\log 1 + \log base}}\]

    3. Simplified32.8

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

    if 2.14368816692774462e121 < re

    1. Initial program 55.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    2. Taylor expanded around inf 7.8

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

    3. Simplified7.8

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.46186942740825347 \cdot 10^{91}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \log \left(\sqrt[3]{base}\right) + 2 \cdot \left(\log base \cdot \log \left(\sqrt[3]{base}\right)\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -3.6142519675909714 \cdot 10^{-198}:\\ \;\;\;\;\frac{\left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(0.0 \cdot 0.0 + {\left(\log base\right)}^{2}\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}\\ \mathbf{elif}\;re \le 1.09305946911323656 \cdot 10^{-271}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 2.14368816692774462 \cdot 10^{121}:\\ \;\;\;\;\frac{\left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(0.0 \cdot 0.0 + {\left(\log base\right)}^{2}\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))