Average Error: 31.1 → 17.0
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.460546366023378 \cdot 10^{+96}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{if}\;re \le 3.3932987648831726 \cdot 10^{+101}:\\ \;\;\;\;\frac{\log \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Split input into 3 regimes

  2. if re < -8.460546366023378e+96

    1. Initial program 49.2

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

    2. Taylor expanded around -inf 8.6

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

    3. Applied simplify8.5

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

    if -8.460546366023378e+96 < re < 3.3932987648831726e+101

    1. Initial program 21.4

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

    3. Applied add-sqr-sqrt21.4

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

    4. Applied sqrt-prod21.4

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

    if 3.3932987648831726e+101 < re

    1. Initial program 51.1

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

    2. Taylor expanded around inf 8.4

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

    3. Applied simplify8.4

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(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))))