Average Error: 31.2 → 18.3
Time: 1.3m
Precision: 64
Internal Precision: 384
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-1}{im} \le -2.8967235600195526 \cdot 10^{+228}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{if}\;\frac{-1}{im} \le -1.8382704077162632 \cdot 10^{+213}:\\ \;\;\;\;\frac{\frac{\left(-2\right) \cdot \frac{1}{2}}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(\frac{-1}{re}\right)}}\\ \mathbf{if}\;\frac{-1}{im} \le -1.7437641998906559 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{{\left(\log \left(im \cdot im + re \cdot re\right)\right)}^{3}}}{\sqrt{\log 10}}\\ \mathbf{if}\;\frac{-1}{im} \le 5.172453989942203 \cdot 10^{-301}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(2 \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{if}\;\frac{-1}{im} \le 1.743144814668755 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\frac{1}{\log 10}} \cdot \left(\log \left(\frac{-1}{im}\right) \cdot \frac{\frac{1}{2} \cdot \sqrt[3]{-8}}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \sqrt[3]{\frac{\log \left(im \cdot im + re \cdot re\right) \cdot \log \left(im \cdot im + re \cdot re\right)}{\frac{\log 10 \cdot \sqrt{\log 10}}{\log \left(im \cdot im + re \cdot re\right)}}}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 6 regimes

  2. if (/ -1 im) < -2.8967235600195526e+228

    1. Initial program 30.8

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

    2. Taylor expanded around inf 35.3

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

    if -2.8967235600195526e+228 < (/ -1 im) < -1.8382704077162632e+213

    1. Initial program 27.1

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

    3. Applied add-sqr-sqrt27.1

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

    4. Applied pow1/227.1

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

    5. Applied log-pow27.1

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

    6. Applied times-frac27.0

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

    8. Applied div-inv26.9

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

    9. Taylor expanded around -inf 32.0

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

    10. Applied simplify32.1

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

    if -1.8382704077162632e+213 < (/ -1 im) < -1.7437641998906559e-62

    1. Initial program 18.5

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

    3. Applied add-sqr-sqrt18.5

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

    4. Applied pow1/218.5

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

    5. Applied log-pow18.5

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

    6. Applied times-frac18.5

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

    8. Applied add-cbrt-cube18.6

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\color{blue}{\sqrt[3]{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \log \left(re \cdot re + im \cdot im\right)}}}{\sqrt{\log 10}}\]

    9. Applied simplify18.6

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{\color{blue}{{\left(\log \left(im \cdot im + re \cdot re\right)\right)}^{3}}}}{\sqrt{\log 10}}\]

    if -1.7437641998906559e-62 < (/ -1 im) < 5.172453989942203e-301

    1. Initial program 46.1

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

    3. Applied add-sqr-sqrt46.1

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

    4. Applied pow1/246.1

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

    5. Applied log-pow46.1

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

    6. Applied times-frac46.1

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

    7. Taylor expanded around 0 12.5

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

    if 5.172453989942203e-301 < (/ -1 im) < 1.743144814668755e-103

    1. Initial program 49.2

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

    3. Applied add-sqr-sqrt49.2

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

    4. Applied pow1/249.2

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

    5. Applied log-pow49.2

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

    6. Applied times-frac49.2

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

    8. Applied add-cbrt-cube49.2

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\color{blue}{\sqrt[3]{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \log \left(re \cdot re + im \cdot im\right)}}}{\sqrt{\log 10}}\]

    9. Applied simplify49.2

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{\color{blue}{{\left(\log \left(im \cdot im + re \cdot re\right)\right)}^{3}}}}{\sqrt{\log 10}}\]

    10. Taylor expanded around -inf 62.0

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\frac{-1}{im}\right) \cdot \left({-8}^{\frac{1}{3}} \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]

    11. Applied simplify8.8

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\log 10}} \cdot \left(\log \left(\frac{-1}{im}\right) \cdot \frac{\frac{1}{2} \cdot \sqrt[3]{-8}}{\sqrt{\log 10}}\right)}\]

    if 1.743144814668755e-103 < (/ -1 im)

    1. Initial program 22.2

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

    3. Applied add-sqr-sqrt22.2

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

    4. Applied pow1/222.2

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

    5. Applied log-pow22.2

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

    6. Applied times-frac22.2

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

    8. Applied add-cbrt-cube22.2

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\color{blue}{\sqrt[3]{\left(\sqrt{\log 10} \cdot \sqrt{\log 10}\right) \cdot \sqrt{\log 10}}}}\]

    9. Applied add-cbrt-cube22.3

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\color{blue}{\sqrt[3]{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \log \left(re \cdot re + im \cdot im\right)}}}{\sqrt[3]{\left(\sqrt{\log 10} \cdot \sqrt{\log 10}\right) \cdot \sqrt{\log 10}}}\]

    10. Applied cbrt-undiv22.2

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\sqrt[3]{\frac{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \log \left(re \cdot re + im \cdot im\right)}{\left(\sqrt{\log 10} \cdot \sqrt{\log 10}\right) \cdot \sqrt{\log 10}}}}\]

    11. Applied simplify22.2

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \sqrt[3]{\color{blue}{\frac{\log \left(im \cdot im + re \cdot re\right) \cdot \log \left(im \cdot im + re \cdot re\right)}{\frac{\log 10 \cdot \sqrt{\log 10}}{\log \left(im \cdot im + re \cdot re\right)}}}}\]
  3. Recombined 6 regimes into one program.

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))