math.log10 on complex, real part

Average Error: 31.1 → 17.7

Time: 14.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.5790703821962546 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(-re\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -3.263660527236801 \cdot 10^{-218}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right) + \log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right)\right) \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) + \sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right)\right)\right)\\ \mathbf{elif}\;re \le 9.76737562426656 \cdot 10^{-213}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \log im\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 2.584078545341751 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\left(\sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

C

double f(double re, double im) {
        double r590994 = re;
        double r590995 = r590994 * r590994;
        double r590996 = im;
        double r590997 = r590996 * r590996;
        double r590998 = r590995 + r590997;
        double r590999 = sqrt(r590998);
        double r591000 = log(r590999);
        double r591001 = 10.0;
        double r591002 = log(r591001);
        double r591003 = r591000 / r591002;
        return r591003;
}

↓

double f(double re, double im) {
        double r591004 = re;
        double r591005 = -1.5790703821962546e+117;
        bool r591006 = r591004 <= r591005;
        double r591007 = 1.0;
        double r591008 = 10.0;
        double r591009 = log(r591008);
        double r591010 = sqrt(r591009);
        double r591011 = r591007 / r591010;
        double r591012 = -r591004;
        double r591013 = log(r591012);
        double r591014 = r591013 * r591011;
        double r591015 = r591011 * r591014;
        double r591016 = -3.263660527236801e-218;
        bool r591017 = r591004 <= r591016;
        double r591018 = im;
        double r591019 = r591018 * r591018;
        double r591020 = r591004 * r591004;
        double r591021 = r591019 + r591020;
        double r591022 = sqrt(r591021);
        double r591023 = cbrt(r591022);
        double r591024 = log(r591023);
        double r591025 = r591024 + r591024;
        double r591026 = sqrt(r591011);
        double r591027 = r591026 * r591026;
        double r591028 = r591025 * r591027;
        double r591029 = r591026 * r591024;
        double r591030 = r591026 * r591029;
        double r591031 = r591028 + r591030;
        double r591032 = r591011 * r591031;
        double r591033 = 9.76737562426656e-213;
        bool r591034 = r591004 <= r591033;
        double r591035 = log(r591018);
        double r591036 = r591011 * r591035;
        double r591037 = r591036 * r591011;
        double r591038 = 2.584078545341751e+99;
        bool r591039 = r591004 <= r591038;
        double r591040 = sqrt(r591026);
        double r591041 = log(r591022);
        double r591042 = r591040 * r591041;
        double r591043 = r591042 * r591040;
        double r591044 = r591043 * r591026;
        double r591045 = r591011 * r591044;
        double r591046 = log(r591004);
        double r591047 = r591011 * r591046;
        double r591048 = r591047 * r591011;
        double r591049 = r591039 ? r591045 : r591048;
        double r591050 = r591034 ? r591037 : r591049;
        double r591051 = r591017 ? r591032 : r591050;
        double r591052 = r591006 ? r591015 : r591051;
        return r591052;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -1.5790703821962546e+117

   1. Initial program 53.0

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

   3. Applied add-sqr-sqrt 53.0

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

   4. Applied pow1 53.0

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

   5. Applied log-pow 53.0

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

   6. Applied times-frac 53.0

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

   8. Applied div-inv 53.0

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

   9. Applied associate-*r* 53.0

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

   10. Taylor expanded around -inf 8.2

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

   11. Simplified 8.2

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

   if -1.5790703821962546e+117 < re < -3.263660527236801e-218

   1. Initial program 18.5

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

   3. Applied add-sqr-sqrt 18.5

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

   4. Applied pow1 18.5

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

   5. Applied log-pow 18.5

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

   6. Applied times-frac 18.5

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

   8. Applied div-inv 18.4

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

   9. Applied associate-*r* 18.4

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

   11. Applied add-sqr-sqrt 18.4

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

   12. Applied associate-*l* 18.5

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

   14. Applied add-cube-cbrt 18.5

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

   15. Applied log-prod 18.5

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

   16. Applied distribute-rgt-in 18.5

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

   17. Applied distribute-lft-in 18.5

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

   18. Simplified 18.4

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

   if -3.263660527236801e-218 < re < 9.76737562426656e-213

   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-sqrt 30.3

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

   4. Applied pow1 30.3

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

   5. Applied log-pow 30.3

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

   6. Applied times-frac 30.3

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

   8. Applied div-inv 30.2

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

   9. Applied associate-*r* 30.2

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

   10. Taylor expanded around 0 33.5

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

   if 9.76737562426656e-213 < re < 2.584078545341751e+99

   1. Initial program 18.9

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

   3. Applied add-sqr-sqrt 18.9

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

   4. Applied pow1 18.9

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

   5. Applied log-pow 18.9

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

   6. Applied times-frac 18.9

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

   8. Applied div-inv 18.8

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

   9. Applied associate-*r* 18.8

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

   11. Applied add-sqr-sqrt 18.8

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

   12. Applied associate-*l* 18.8

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

   14. Applied add-sqr-sqrt 18.8

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

   15. Applied sqrt-prod 18.9

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

   16. Applied associate-*l* 18.9

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

   if 2.584078545341751e+99 < re

   1. Initial program 50.5

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

   3. Applied add-sqr-sqrt 50.5

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

   4. Applied pow1 50.5

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

   5. Applied log-pow 50.5

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

   6. Applied times-frac 50.5

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

   8. Applied div-inv 50.5

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

   9. Applied associate-*r* 50.5

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

   10. Taylor expanded around inf 9.1

       \[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{re}\right) \cdot \frac{1}{\sqrt{\log 10}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.5790703821962546 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(-re\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -3.263660527236801 \cdot 10^{-218}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right) + \log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right)\right) \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) + \sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \log \left(\sqrt[3]{\sqrt{im \cdot im + re \cdot re}}\right)\right)\right)\\ \mathbf{elif}\;re \le 9.76737562426656 \cdot 10^{-213}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \log im\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 2.584078545341751 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\left(\sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\log 10}} \cdot \log re\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))