Average Error: 32.0 → 18.8
Time: 46.2s
Precision: 64
Internal Precision: 384
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.130257577176994 \cdot 10^{+140}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{if}\;re \le -3.316585159322524 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{if}\;re \le 2.375186144680828 \cdot 10^{-274}:\\ \;\;\;\;\frac{\log \left(-im\right)}{\log 10}\\ \mathbf{if}\;re \le 1.1347647040549314 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{if}\;re \le 6.107346154734439 \cdot 10^{-166}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \mathbf{if}\;re \le 5.72970788763429 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -2.130257577176994e+140

    1. Initial program 58.2

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

    2. Taylor expanded around -inf 7.5

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

    3. Applied simplify7.5

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

    if -2.130257577176994e+140 < re < -3.316585159322524e-245 or 2.375186144680828e-274 < re < 1.1347647040549314e-245 or 6.107346154734439e-166 < re < 5.72970788763429e+146

    1. Initial program 19.0

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

    3. Applied add-sqr-sqrt19.0

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

    4. Applied pow1/219.0

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

    5. Applied log-pow19.0

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

    6. Applied times-frac19.0

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

    8. Applied div-inv18.9

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

    if -3.316585159322524e-245 < re < 2.375186144680828e-274

    1. Initial program 29.7

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

    3. Applied add-cbrt-cube30.1

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

    4. Applied add-cbrt-cube30.0

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

    5. Applied cbrt-undiv29.7

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

    6. Applied simplify29.7

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

    7. Taylor expanded around -inf 33.1

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

    8. Applied simplify33.1

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

    if 1.1347647040549314e-245 < re < 6.107346154734439e-166 or 5.72970788763429e+146 < re

    1. Initial program 51.5

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

    2. Taylor expanded around inf 20.7

      \[\leadsto \frac{\log \color{blue}{re}}{\log 10}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 46.2s)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))