Average Error: 31.7 → 17.7
Time: 19.7s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.47409571178928762 \cdot 10^{117}:\\ \;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le 5.43513758536357538 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.47409571178928762 \cdot 10^{117}:\\
\;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\

\mathbf{elif}\;re \le 5.43513758536357538 \cdot 10^{84}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}{\sqrt{\log 10}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\

\end{array}
double f(double re, double im) {
        double r37175 = re;
        double r37176 = r37175 * r37175;
        double r37177 = im;
        double r37178 = r37177 * r37177;
        double r37179 = r37176 + r37178;
        double r37180 = sqrt(r37179);
        double r37181 = log(r37180);
        double r37182 = 10.0;
        double r37183 = log(r37182);
        double r37184 = r37181 / r37183;
        return r37184;
}
↓
double f(double re, double im) {
        double r37185 = re;
        double r37186 = -2.4740957117892876e+117;
        bool r37187 = r37185 <= r37186;
        double r37188 = -1.0;
        double r37189 = 10.0;
        double r37190 = log(r37189);
        double r37191 = sqrt(r37190);
        double r37192 = r37188 / r37191;
        double r37193 = r37188 / r37185;
        double r37194 = log(r37193);
        double r37195 = 1.0;
        double r37196 = r37195 / r37190;
        double r37197 = sqrt(r37196);
        double r37198 = r37194 * r37197;
        double r37199 = r37192 * r37198;
        double r37200 = 5.435137585363575e+84;
        bool r37201 = r37185 <= r37200;
        double r37202 = r37195 / r37191;
        double r37203 = r37185 * r37185;
        double r37204 = im;
        double r37205 = r37204 * r37204;
        double r37206 = r37203 + r37205;
        double r37207 = sqrt(r37206);
        double r37208 = sqrt(r37207);
        double r37209 = r37208 * r37208;
        double r37210 = log(r37209);
        double r37211 = r37210 / r37191;
        double r37212 = r37202 * r37211;
        double r37213 = log(r37185);
        double r37214 = r37197 * r37213;
        double r37215 = r37202 * r37214;
        double r37216 = r37201 ? r37212 : r37215;
        double r37217 = r37187 ? r37199 : r37216;
        return r37217;
}

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 < -2.4740957117892876e+117

    1. Initial program 55.5

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

    3. Applied add-sqr-sqrt55.5

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

    4. Applied pow155.5

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

    5. Applied log-pow55.5

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

    6. Applied times-frac55.5

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

    7. Taylor expanded around -inf 8.3

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

    8. Simplified8.3

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

    if -2.4740957117892876e+117 < re < 5.435137585363575e+84

    1. Initial program 22.0

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

    3. Applied add-sqr-sqrt22.0

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

    4. Applied pow122.0

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

    5. Applied log-pow22.0

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

    6. Applied times-frac21.9

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

    8. Applied add-sqr-sqrt21.9

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

    9. Applied sqrt-prod21.9

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

    if 5.435137585363575e+84 < re

    1. Initial program 48.0

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

    3. Applied add-sqr-sqrt48.0

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

    4. Applied pow148.0

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

    5. Applied log-pow48.0

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

    6. Applied times-frac48.0

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

    8. Applied add-sqr-sqrt48.0

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

    9. Applied sqrt-prod48.0

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

    10. Taylor expanded around inf 9.8

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

    11. Simplified9.8

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.47409571178928762 \cdot 10^{117}:\\ \;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le 5.43513758536357538 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))