Average Error: 31.3 → 17.3
Time: 44.8s
Precision: 64
Internal Precision: 576
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.4509602360394443 \cdot 10^{+61}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{if}\;re \le 3.8235481247063786 \cdot 10^{+117}:\\ \;\;\;\;\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot e^{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

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 < -2.4509602360394443e+61

    1. Initial program 45.5

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

    2. Taylor expanded around -inf 10.1

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

    3. Applied simplify10.1

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

    if -2.4509602360394443e+61 < re < 3.8235481247063786e+117

    1. Initial program 21.5

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

    3. Applied add-cube-cbrt21.5

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

    5. Applied add-exp-log21.5

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

    if 3.8235481247063786e+117 < re

    1. Initial program 53.1

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

    2. Taylor expanded around inf 9.1

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

Runtime

Time bar (total: 44.8s)Debug logProfile

herbie shell --seed 2020178 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))