math.log10 on complex, real part

Average Error: 32.1 → 19.2

Time: 9.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -1.9204131668484536 \cdot 10^{47}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\log \left(\frac{-1}{im}\right) \cdot -2\right)}{\log 10}\\ \mathbf{elif}\;im \le 9.50561434604941541 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\sqrt{\frac{1}{2}}\right)}\right)}{\log 10}\\ \mathbf{elif}\;im \le 5.747339276714826 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\left(-\log re\right) \cdot -2\right)\right)\\ \mathbf{elif}\;im \le 1.7222444720608974 \cdot 10^{37}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\sqrt{\frac{1}{2}}\right)}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(2 \cdot \log im\right)\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if im < -1.9204131668484536e47

   1. Initial program 44.5

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

   3. Applied add-sqr-sqrt 44.5

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

   4. Applied pow1/2 44.5

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 44.5

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 44.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 44.4

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{1 \cdot \sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]

   9. Applied add-sqr-sqrt 44.5

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{1 \cdot \sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]

   10. Applied times-frac 44.4

       \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{1}{2}}}{1} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]

   11. Applied associate-*l* 44.4

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{1} \cdot \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)}\]

   12. Simplified 44.4

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{1} \cdot \color{blue}{\frac{\sqrt{\frac{1}{2}} \cdot \log \left(im \cdot im + re \cdot re\right)}{\log 10}}\]

   13. Taylor expanded around -inf 10.9

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{1} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \color{blue}{\left(\log 1 - 2 \cdot \log \left(\frac{-1}{im}\right)\right)}}{\log 10}\]

   14. Simplified 10.9

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{1} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \color{blue}{\left(\log \left(\frac{-1}{im}\right) \cdot -2\right)}}{\log 10}\]

   if -1.9204131668484536e47 < im < 9.50561434604941541e-245 or 5.747339276714826e-102 < im < 1.7222444720608974e37

   1. Initial program 21.9

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

   3. Applied add-sqr-sqrt 21.9

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

   4. Applied pow1/2 21.9

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 21.9

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 21.9

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 21.9

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{1 \cdot \sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]

   9. Applied add-sqr-sqrt 21.9

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{1 \cdot \sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]

   10. Applied times-frac 21.9

       \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{1}{2}}}{1} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]

   11. Applied associate-*l* 21.8

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{1} \cdot \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)}\]

   12. Simplified 21.8

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{1} \cdot \color{blue}{\frac{\sqrt{\frac{1}{2}} \cdot \log \left(im \cdot im + re \cdot re\right)}{\log 10}}\]
   13. Using strategy rm

   14. Applied add-log-exp 21.8

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{1} \cdot \frac{\color{blue}{\log \left(e^{\sqrt{\frac{1}{2}} \cdot \log \left(im \cdot im + re \cdot re\right)}\right)}}{\log 10}\]

   15. Simplified 21.8

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{1} \cdot \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\left(\sqrt{\frac{1}{2}}\right)}\right)}}{\log 10}\]

   if 9.50561434604941541e-245 < im < 5.747339276714826e-102

   1. Initial program 26.6

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

   3. Applied add-sqr-sqrt 26.6

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

   4. Applied pow1/2 26.6

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 26.6

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 26.6

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]

   7. Taylor expanded around inf 39.5

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{re}\right)\right)\right)}\]

   8. Simplified 39.5

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(\left(-\log re\right) \cdot -2\right)\right)}\]

   if 1.7222444720608974e37 < im

   1. Initial program 44.1

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

   3. Applied add-sqr-sqrt 44.1

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

   4. Applied pow1/2 44.1

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 44.1

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 44.1

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]

   7. Taylor expanded around 0 11.4

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log 1 + 2 \cdot \log im\right)\right)}\]

   8. Simplified 11.4

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\log 10}} \cdot \left(2 \cdot \log im\right)\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 19.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.9204131668484536 \cdot 10^{47}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\log \left(\frac{-1}{im}\right) \cdot -2\right)}{\log 10}\\ \mathbf{elif}\;im \le 9.50561434604941541 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\sqrt{\frac{1}{2}}\right)}\right)}{\log 10}\\ \mathbf{elif}\;im \le 5.747339276714826 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\left(-\log re\right) \cdot -2\right)\right)\\ \mathbf{elif}\;im \le 1.7222444720608974 \cdot 10^{37}:\\ \;\;\;\;\sqrt{\frac{1}{2}} \cdot \frac{\log \left({\left(re \cdot re + im \cdot im\right)}^{\left(\sqrt{\frac{1}{2}}\right)}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(2 \cdot \log im\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))