Average Error: 30.5 → 18.3
Time: 52.8s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.7671689040102433 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le -3.8651586381031656 \cdot 10^{-177}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\sqrt{\log 10}}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{elif}\;re \le -2.4458756963213023 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le 1.0074545562893787 \cdot 10^{-220}:\\ \;\;\;\;\left(\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10}}}}\right) \cdot \sqrt{\frac{\sqrt{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|}}\\ \mathbf{elif}\;re \le 1.2586540654634947 \cdot 10^{-190}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log re \cdot 2\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.3957084041509018 \cdot 10^{+84}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\sqrt{\log 10}}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log re \cdot 2\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -3.7671689040102433e+133 or -3.8651586381031656e-177 < re < -2.4458756963213023e-235

    1. Initial program 49.1

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

    3. Applied add-sqr-sqrt49.1

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

    4. Applied pow1/249.1

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

    5. Applied log-pow49.1

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

    6. Applied times-frac49.1

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

    7. Taylor expanded around -inf 18.7

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

    if -3.7671689040102433e+133 < re < -3.8651586381031656e-177 or 1.2586540654634947e-190 < re < 1.3957084041509018e+84

    1. Initial program 17.1

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

    3. Applied add-sqr-sqrt17.1

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

    4. Applied pow1/217.1

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

    5. Applied log-pow17.1

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

    6. Applied times-frac17.1

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

    8. Applied add-sqr-sqrt17.1

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

    9. Applied associate-*l*17.0

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

    11. Applied associate-*r/17.0

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

    13. Applied add-sqr-sqrt17.0

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

    14. Applied sqrt-prod17.2

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

    15. Applied times-frac17.1

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

    16. Applied associate-*r*17.0

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

    if -2.4458756963213023e-235 < re < 1.0074545562893787e-220

    1. Initial program 30.3

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

    3. Applied add-sqr-sqrt30.3

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

    4. Applied pow1/230.3

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

    5. Applied log-pow30.3

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

    6. Applied times-frac30.3

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

    8. Applied add-sqr-sqrt30.3

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

    9. Applied associate-*l*30.2

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

    11. Applied add-cube-cbrt30.5

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

    12. Applied sqrt-prod30.5

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

    13. Applied add-sqr-sqrt30.2

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

    14. Applied times-frac30.5

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

    15. Applied sqrt-prod30.2

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

    16. Applied associate-*l*30.2

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

    17. Simplified30.2

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

    if 1.0074545562893787e-220 < re < 1.2586540654634947e-190 or 1.3957084041509018e+84 < re

    1. Initial program 44.4

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

    3. Applied add-sqr-sqrt44.4

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

    4. Applied pow1/244.4

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

    5. Applied log-pow44.4

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

    6. Applied times-frac44.4

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

    7. Taylor expanded around inf 13.5

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

    8. Simplified13.5

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

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.7671689040102433 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le -3.8651586381031656 \cdot 10^{-177}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\sqrt{\log 10}}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{elif}\;re \le -2.4458756963213023 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le 1.0074545562893787 \cdot 10^{-220}:\\ \;\;\;\;\left(\left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10}}}}\right) \cdot \sqrt{\frac{\sqrt{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|}}\\ \mathbf{elif}\;re \le 1.2586540654634947 \cdot 10^{-190}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log re \cdot 2\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.3957084041509018 \cdot 10^{+84}:\\ \;\;\;\;\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\sqrt{\sqrt{\log 10}}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log re \cdot 2\right)\right) \cdot \frac{\frac{1}{2}}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))