math.log/2 on complex, real part

Average Error: 32.1 → 17.8

Time: 17.9s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.0752208244632336 \cdot 10^{60}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -2.88208828919163744 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 4.7115991876438505 \cdot 10^{-204}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 6.31625123356129944 \cdot 10^{37}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 4 regimes

2. if re < -2.0752208244632336e60

   1. Initial program 45.2

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

   2. Taylor expanded around -inf 10.5

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

   3. Simplified 10.5

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

   if -2.0752208244632336e60 < re < -2.88208828919163744e-308 or 4.7115991876438505e-204 < re < 6.31625123356129944e37

   1. Initial program 21.2

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

   3. Applied add-sqr-sqrt 21.2

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

   4. Applied *-un-lft-identity 21.2

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

   5. Applied times-frac 21.2

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

   6. Simplified 21.2

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

   7. Simplified 21.2

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

   if -2.88208828919163744e-308 < re < 4.7115991876438505e-204

   1. Initial program 33.2

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

   2. Taylor expanded around 0 33.8

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

   3. Simplified 33.8

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

   if 6.31625123356129944e37 < re

   1. Initial program 43.1

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

   2. Taylor expanded around inf 11.3

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

   3. Simplified 11.3

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.0752208244632336 \cdot 10^{60}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -2.88208828919163744 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 4.7115991876438505 \cdot 10^{-204}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 6.31625123356129944 \cdot 10^{37}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))