math.log10 on complex, real part

Average Error: 31.0 → 17.1

Time: 40.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.103637907474162 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(\frac{\log \left(\frac{-1}{re}\right) \cdot -2}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10}}}}\right) \cdot \sqrt{\frac{\sqrt{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|}}\\ \mathbf{elif}\;re \le 1.8737997944449135 \cdot 10^{+98}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\log \left(im \cdot im + re \cdot re\right)}{\sqrt{\log 10}}\right) \cdot \sqrt{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2 \cdot \log re}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \end{array}\]

C

double f(double re, double im) {
        double r1685099 = re;
        double r1685100 = r1685099 * r1685099;
        double r1685101 = im;
        double r1685102 = r1685101 * r1685101;
        double r1685103 = r1685100 + r1685102;
        double r1685104 = sqrt(r1685103);
        double r1685105 = log(r1685104);
        double r1685106 = 10.0;
        double r1685107 = log(r1685106);
        double r1685108 = r1685105 / r1685107;
        return r1685108;
}

↓

double f(double re, double im) {
        double r1685109 = re;
        double r1685110 = -2.103637907474162e+73;
        bool r1685111 = r1685109 <= r1685110;
        double r1685112 = -1.0;
        double r1685113 = r1685112 / r1685109;
        double r1685114 = log(r1685113);
        double r1685115 = -2.0;
        double r1685116 = r1685114 * r1685115;
        double r1685117 = 10.0;
        double r1685118 = log(r1685117);
        double r1685119 = sqrt(r1685118);
        double r1685120 = r1685116 / r1685119;
        double r1685121 = 0.5;
        double r1685122 = r1685121 / r1685119;
        double r1685123 = sqrt(r1685122);
        double r1685124 = r1685120 * r1685123;
        double r1685125 = sqrt(r1685121);
        double r1685126 = cbrt(r1685118);
        double r1685127 = sqrt(r1685126);
        double r1685128 = r1685125 / r1685127;
        double r1685129 = sqrt(r1685128);
        double r1685130 = r1685124 * r1685129;
        double r1685131 = fabs(r1685126);
        double r1685132 = r1685125 / r1685131;
        double r1685133 = sqrt(r1685132);
        double r1685134 = r1685130 * r1685133;
        double r1685135 = 1.8737997944449135e+98;
        bool r1685136 = r1685109 <= r1685135;
        double r1685137 = r1685125 / r1685119;
        double r1685138 = im;
        double r1685139 = r1685138 * r1685138;
        double r1685140 = r1685109 * r1685109;
        double r1685141 = r1685139 + r1685140;
        double r1685142 = log(r1685141);
        double r1685143 = r1685142 / r1685119;
        double r1685144 = r1685137 * r1685143;
        double r1685145 = r1685144 * r1685125;
        double r1685146 = 2.0;
        double r1685147 = log(r1685109);
        double r1685148 = r1685146 * r1685147;
        double r1685149 = r1685148 / r1685119;
        double r1685150 = r1685149 * r1685123;
        double r1685151 = r1685150 * r1685123;
        double r1685152 = r1685136 ? r1685145 : r1685151;
        double r1685153 = r1685111 ? r1685134 : r1685152;
        return r1685153;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -2.103637907474162e+73

   1. Initial program 46.2

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

   3. Applied add-sqr-sqrt 46.2

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

   4. Applied pow1 46.2

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

   5. Applied sqrt-pow1 46.2

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

   6. Applied log-pow 46.2

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

   7. Applied times-frac 46.2

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

   8. Simplified 46.2

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

   10. Applied add-sqr-sqrt 46.2

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

   11. Applied associate-*l* 46.2

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

   13. Applied add-cube-cbrt 46.3

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

   14. Applied sqrt-prod 46.3

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

   15. Applied add-sqr-sqrt 46.2

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

   16. Applied times-frac 46.3

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

   17. Applied sqrt-prod 46.2

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

   18. Applied associate-*l* 46.2

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

   19. Simplified 46.2

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

   20. Taylor expanded around -inf 10.1

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

   21. Simplified 10.1

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

   if -2.103637907474162e+73 < re < 1.8737997944449135e+98

   1. Initial program 21.3

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

   3. Applied add-sqr-sqrt 21.3

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

   4. Applied pow1 21.3

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

   5. Applied sqrt-pow1 21.3

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

   6. Applied log-pow 21.3

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

   7. Applied times-frac 21.3

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

   8. Simplified 21.3

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

   10. Applied *-un-lft-identity 21.3

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

   11. Applied sqrt-prod 21.3

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

   12. Applied add-sqr-sqrt 21.3

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

   13. Applied times-frac 21.3

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

   14. Applied associate-*l* 21.2

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

   15. Simplified 21.2

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

   if 1.8737997944449135e+98 < re

   1. Initial program 49.3

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

   3. Applied add-sqr-sqrt 49.3

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

   4. Applied pow1 49.3

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

   5. Applied sqrt-pow1 49.3

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

   6. Applied log-pow 49.3

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

   7. Applied times-frac 49.3

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

   8. Simplified 49.3

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

   10. Applied add-sqr-sqrt 49.3

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

   11. Applied associate-*l* 49.2

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

   12. Taylor expanded around inf 10.0

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

   13. Simplified 10.0

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.103637907474162 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(\frac{\log \left(\frac{-1}{re}\right) \cdot -2}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt[3]{\log 10}}}}\right) \cdot \sqrt{\frac{\sqrt{\frac{1}{2}}}{\left|\sqrt[3]{\log 10}\right|}}\\ \mathbf{elif}\;re \le 1.8737997944449135 \cdot 10^{+98}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\log \left(im \cdot im + re \cdot re\right)}{\sqrt{\log 10}}\right) \cdot \sqrt{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2 \cdot \log re}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))