Average Error: 31.5 → 17.3
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{re} \le -1.3818739401625216 \cdot 10^{-99}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{if}\;\frac{1}{re} \le 4.868412342987585 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(-2 \cdot \left(\left(\sqrt{\frac{1}{2}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}}\right)\right)\\ \mathbf{if}\;\frac{1}{re} \le 1.3643519944597905 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log re \cdot \left(--2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 re) < -1.3818739401625216e-99 or 1.3643519944597905e-78 < (/ 1 re)

    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-sqrt21.9

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

    4. Applied pow1/221.9

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

    5. Applied log-pow21.9

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

    6. Applied times-frac21.8

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

    8. Applied add-sqr-sqrt21.8

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

    9. Applied associate-*l*21.7

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

    if -1.3818739401625216e-99 < (/ 1 re) < 4.868412342987585e-308

    1. Initial program 49.8

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

    3. Applied add-sqr-sqrt49.8

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

    4. Applied pow1/249.8

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

    5. Applied log-pow49.8

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

    6. Applied times-frac49.8

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

    8. Applied add-sqr-sqrt49.8

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

    9. Applied associate-*l*49.8

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

    10. Taylor expanded around -inf 8.9

      \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \color{blue}{\left(-2 \cdot \left(\left(\sqrt{\frac{1}{2}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot {\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}}\right)\right)}\]

    if 4.868412342987585e-308 < (/ 1 re) < 1.3643519944597905e-78

    1. Initial program 46.4

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

    3. Applied add-sqr-sqrt46.4

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

    4. Applied pow1/246.4

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

    5. Applied log-pow46.4

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

    6. Applied times-frac46.4

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

    7. Taylor expanded around inf 10.4

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

    8. Applied simplify10.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))