math.log10 on complex, real part

Average Error: 32.2 → 18.7

Time: 8.0s

Precision: binary64

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.21504708687891476 \cdot 10^{144}:\\ \;\;\;\;-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le -4.3407838631968708 \cdot 10^{-187}:\\ \;\;\;\;\frac{3}{\frac{1}{3} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}\\ \mathbf{elif}\;re \le -2.30737807072558918 \cdot 10^{-294}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{im}\right)}}\\ \mathbf{elif}\;re \le 1.60770801328417031 \cdot 10^{-255}:\\ \;\;\;\;\frac{3}{\frac{1}{3} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}\\ \mathbf{elif}\;re \le 2.676988169724194 \cdot 10^{-193}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \le 1.29362631726006162 \cdot 10^{74}:\\ \;\;\;\;\frac{3}{\frac{1}{3} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{re}\right)}}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -1.21504708687891476e144

   1. Initial program 61.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

   2. Taylor expanded around -inf 8.6

      \[\leadsto \color{blue}{\frac{\log 1 - \log \left(\frac{-1}{re}\right)}{\log 10}}\]

   3. Simplified 8.6

      \[\leadsto \color{blue}{-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}}\]

   if -1.21504708687891476e144 < re < -4.3407838631968708e-187 or -2.30737807072558918e-294 < re < 1.60770801328417031e-255 or 2.676988169724194e-193 < re < 1.29362631726006162e74

   1. Initial program 19.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 19.7

      \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}{\log 10}\]
   4. Using strategy rm

   5. Applied pow3 19.7

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{3}\right)}}{\log 10}\]

   6. Applied log-pow 19.7

      \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}{\log 10}\]

   7. Applied associate-/l* 19.7

      \[\leadsto \color{blue}{\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 19.7

      \[\leadsto \frac{3}{\frac{\log 10}{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}}\]
   10. Using strategy rm

   11. Applied pow1 19.7

       \[\leadsto \frac{3}{\frac{\log 10}{\log \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right) \cdot \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}^{1}}\right)}}\]

   12. Applied pow1 19.7

       \[\leadsto \frac{3}{\frac{\log 10}{\log \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}^{1}\right)}}\]

   13. Applied pow1 19.7

       \[\leadsto \frac{3}{\frac{\log 10}{\log \left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}^{1}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}^{1}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}^{1}\right)}}\]

   14. Applied pow-prod-up 19.7

       \[\leadsto \frac{3}{\frac{\log 10}{\log \left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}^{\left(1 + 1\right)}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}^{1}\right)}}\]

   15. Applied pow-prod-up 19.7

       \[\leadsto \frac{3}{\frac{\log 10}{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}^{\left(\left(1 + 1\right) + 1\right)}\right)}}}\]

   16. Applied log-pow 19.7

       \[\leadsto \frac{3}{\frac{\log 10}{\color{blue}{\left(\left(1 + 1\right) + 1\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}}\]

   17. Applied pow1 19.7

       \[\leadsto \frac{3}{\frac{\log \color{blue}{\left({10}^{1}\right)}}{\left(\left(1 + 1\right) + 1\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}\]

   18. Applied log-pow 19.7

       \[\leadsto \frac{3}{\frac{\color{blue}{1 \cdot \log 10}}{\left(\left(1 + 1\right) + 1\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}\]

   19. Applied times-frac 19.6

       \[\leadsto \frac{3}{\color{blue}{\frac{1}{\left(1 + 1\right) + 1} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}}\]

   20. Simplified 19.6

       \[\leadsto \frac{3}{\color{blue}{\frac{1}{3}} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}\]

   if -4.3407838631968708e-187 < re < -2.30737807072558918e-294

   1. Initial program 32.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 32.9

      \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}{\log 10}\]
   4. Using strategy rm

   5. Applied pow3 32.9

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{3}\right)}}{\log 10}\]

   6. Applied log-pow 32.9

      \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}{\log 10}\]

   7. Applied associate-/l* 32.9

      \[\leadsto \color{blue}{\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}}\]

   8. Taylor expanded around 0 35.9

      \[\leadsto \frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\color{blue}{im}}\right)}}\]

   if 1.60770801328417031e-255 < re < 2.676988169724194e-193

   1. Initial program 32.6

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

   2. Taylor expanded around 0 34.8

      \[\leadsto \frac{\log \color{blue}{im}}{\log 10}\]

   if 1.29362631726006162e74 < re

   1. Initial program 47.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 47.4

      \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}{\log 10}\]
   4. Using strategy rm

   5. Applied pow3 47.4

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}^{3}\right)}}{\log 10}\]

   6. Applied log-pow 47.4

      \[\leadsto \frac{\color{blue}{3 \cdot \log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}{\log 10}\]

   7. Applied associate-/l* 47.4

      \[\leadsto \color{blue}{\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}}\]

   8. Taylor expanded around inf 11.5

      \[\leadsto \frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{\color{blue}{re}}\right)}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.21504708687891476 \cdot 10^{144}:\\ \;\;\;\;-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le -4.3407838631968708 \cdot 10^{-187}:\\ \;\;\;\;\frac{3}{\frac{1}{3} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}\\ \mathbf{elif}\;re \le -2.30737807072558918 \cdot 10^{-294}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{im}\right)}}\\ \mathbf{elif}\;re \le 1.60770801328417031 \cdot 10^{-255}:\\ \;\;\;\;\frac{3}{\frac{1}{3} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}\\ \mathbf{elif}\;re \le 2.676988169724194 \cdot 10^{-193}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \le 1.29362631726006162 \cdot 10^{74}:\\ \;\;\;\;\frac{3}{\frac{1}{3} \cdot \frac{\log 10}{\log \left(\sqrt[3]{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{re}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))