Average Error: 30.7 → 16.7
Time: 1.8m
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;-im \le -4.8018804759720435 \cdot 10^{+145}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{if}\;-im \le 1.0819036470003008 \cdot 10^{+117}:\\ \;\;\;\;\frac{\log \left(\left|\sqrt[3]{im \cdot im + re \cdot re}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log base}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Split input into 3 regimes

  2. if (- im) < -4.8018804759720435e+145

    1. Initial program 59.8

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

    2. Taylor expanded around 0 7.7

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

    if -4.8018804759720435e+145 < (- im) < 1.0819036470003008e+117

    1. Initial program 20.2

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

    3. Applied add-cube-cbrt20.2

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

    4. Applied sqrt-prod20.2

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

    5. Applied simplify20.2

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

    if 1.0819036470003008e+117 < (- 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. Using strategy rm

    3. Applied clear-num53.7

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

    4. Applied simplify53.7

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

    5. Taylor expanded around -inf 8.3

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

    6. Applied simplify8.2

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(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))))