Average Error: 30.4 → 17.1
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-1}{re} \le -1.3576420698193105 \cdot 10^{-95}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}\\ \mathbf{if}\;\frac{-1}{re} \le 3.928193180907568 \cdot 10^{-306}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\log re \cdot \left(--2\right)\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{if}\;\frac{-1}{re} \le 2.27589279457223 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}{\log 10}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if (/ -1 re) < -1.3576420698193105e-95

    1. Initial program 20.9

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

    3. Applied add-cbrt-cube21.5

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

    4. Applied add-cbrt-cube21.4

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

    5. Applied cbrt-undiv21.0

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

    6. Applied simplify21.0

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

    if -1.3576420698193105e-95 < (/ -1 re) < 3.928193180907568e-306

    1. Initial program 48.7

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

    3. Applied add-sqr-sqrt48.7

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

    4. Applied pow1/248.7

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

    5. Applied log-pow48.7

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

    6. Applied times-frac48.7

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

    7. Taylor expanded around inf 9.6

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

    8. Applied simplify9.6

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

    if 3.928193180907568e-306 < (/ -1 re) < 2.27589279457223e-39

    1. Initial program 41.2

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

    3. Applied add-sqr-sqrt41.2

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

    4. Applied pow1/241.2

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

    5. Applied log-pow41.2

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

    6. Applied times-frac41.2

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

    7. Taylor expanded around -inf 12.2

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

    if 2.27589279457223e-39 < (/ -1 re)

    1. Initial program 21.3

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

    3. Applied add-cube-cbrt21.3

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

    4. Applied sqrt-prod21.3

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

    5. Applied simplify21.3

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018296 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))