Average Error: 31.0 → 17.1
Time: 27.7s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.152998653485218 \cdot 10^{+129}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \le 4.3386027320831174 \cdot 10^{+108}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(im \cdot im + re \cdot re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

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 3 regimes

  2. if re < -1.152998653485218e+129

    1. Initial program 56.0

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

    2. Taylor expanded around -inf 7.1

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

    3. Simplified7.1

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

    if -1.152998653485218e+129 < re < 4.3386027320831174e+108

    1. Initial program 21.2

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

    3. Applied pow1/221.2

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

    4. Applied log-pow21.2

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

    5. Applied associate-/l*21.2

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

    7. Applied *-un-lft-identity21.2

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

    8. Applied add-sqr-sqrt21.2

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

    9. Applied times-frac21.3

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

    10. Applied associate-/r*21.1

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

    if 4.3386027320831174e+108 < re

    1. Initial program 49.9

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

    2. Taylor expanded around inf 9.1

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

    3. Simplified9.1

      \[\leadsto \color{blue}{\frac{\log re}{\log 10}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.152998653485218 \cdot 10^{+129}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{elif}\;re \le 4.3386027320831174 \cdot 10^{+108}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(im \cdot im + re \cdot re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Runtime

Time bar (total: 27.7s)Debug logProfile

herbie shell --seed 2018248 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))