math.log10 on complex, real part

Average Error: 32.8 → 18.0

Time: 13.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.1975508038006968 \cdot 10^{153}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le -4.5405452937718227 \cdot 10^{-275}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \le 1.27662858127337166 \cdot 10^{-281}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt{\sqrt{\frac{1}{2}}}}{\sqrt{\log 10}} \cdot \left(\sqrt{\sqrt{\frac{1}{2}}} \cdot \left(2 \cdot \log im\right)\right)\right)\\ \mathbf{elif}\;re \le 8.0278279982437549 \cdot 10^{98}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{2 \cdot \log re}}\\ \end{array}\]

C

double f(double re, double im) {
        double r46096 = re;
        double r46097 = r46096 * r46096;
        double r46098 = im;
        double r46099 = r46098 * r46098;
        double r46100 = r46097 + r46099;
        double r46101 = sqrt(r46100);
        double r46102 = log(r46101);
        double r46103 = 10.0;
        double r46104 = log(r46103);
        double r46105 = r46102 / r46104;
        return r46105;
}

↓

double f(double re, double im) {
        double r46106 = re;
        double r46107 = -4.197550803800697e+153;
        bool r46108 = r46106 <= r46107;
        double r46109 = -1.0;
        double r46110 = r46109 / r46106;
        double r46111 = log(r46110);
        double r46112 = -r46111;
        double r46113 = 10.0;
        double r46114 = log(r46113);
        double r46115 = r46112 / r46114;
        double r46116 = -4.540545293771823e-275;
        bool r46117 = r46106 <= r46116;
        double r46118 = 0.5;
        double r46119 = sqrt(r46118);
        double r46120 = sqrt(r46114);
        double r46121 = r46119 / r46120;
        double r46122 = r46106 * r46106;
        double r46123 = im;
        double r46124 = r46123 * r46123;
        double r46125 = r46122 + r46124;
        double r46126 = log(r46125);
        double r46127 = r46120 / r46126;
        double r46128 = r46119 / r46127;
        double r46129 = r46121 * r46128;
        double r46130 = 1.2766285812733717e-281;
        bool r46131 = r46106 <= r46130;
        double r46132 = sqrt(r46119);
        double r46133 = r46132 / r46120;
        double r46134 = 2.0;
        double r46135 = log(r46123);
        double r46136 = r46134 * r46135;
        double r46137 = r46132 * r46136;
        double r46138 = r46133 * r46137;
        double r46139 = r46121 * r46138;
        double r46140 = 8.027827998243755e+98;
        bool r46141 = r46106 <= r46140;
        double r46142 = log(r46106);
        double r46143 = r46134 * r46142;
        double r46144 = r46120 / r46143;
        double r46145 = r46119 / r46144;
        double r46146 = r46121 * r46145;
        double r46147 = r46141 ? r46129 : r46146;
        double r46148 = r46131 ? r46139 : r46147;
        double r46149 = r46117 ? r46129 : r46148;
        double r46150 = r46108 ? r46115 : r46149;
        return r46150;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -4.197550803800697e+153

   1. Initial program 63.9

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

   3. Applied pow1 63.9

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

   4. Applied sqrt-pow1 63.9

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

   5. Applied log-pow 63.9

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

   6. Applied associate-/l* 63.9

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

   8. Applied pow1 63.9

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

   9. Applied log-pow 63.9

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

   10. Applied add-sqr-sqrt 63.9

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

   11. Applied times-frac 63.9

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

   12. Applied add-sqr-sqrt 63.9

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

   13. Applied times-frac 63.9

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

   14. Simplified 63.9

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

   15. Simplified 63.9

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

   16. Taylor expanded around -inf 7.2

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

   17. Simplified 7.1

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

   if -4.197550803800697e+153 < re < -4.540545293771823e-275 or 1.2766285812733717e-281 < re < 8.027827998243755e+98

   1. Initial program 21.6

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

   3. Applied pow1 21.6

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

   4. Applied sqrt-pow1 21.6

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

   5. Applied log-pow 21.6

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

   6. Applied associate-/l* 21.7

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

   8. Applied pow1 21.7

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

   9. Applied log-pow 21.7

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

   10. Applied add-sqr-sqrt 21.7

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

   11. Applied times-frac 21.8

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

   12. Applied add-sqr-sqrt 21.7

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

   13. Applied times-frac 21.5

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

   14. Simplified 21.5

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

   15. Simplified 21.5

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

   if -4.540545293771823e-275 < re < 1.2766285812733717e-281

   1. Initial program 33.3

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

   3. Applied pow1 33.3

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

   4. Applied sqrt-pow1 33.3

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

   5. Applied log-pow 33.3

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

   6. Applied associate-/l* 33.2

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

   8. Applied pow1 33.2

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

   9. Applied log-pow 33.2

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

   10. Applied add-sqr-sqrt 33.2

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

   11. Applied times-frac 33.3

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

   12. Applied add-sqr-sqrt 33.3

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

   13. Applied times-frac 33.2

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

   14. Simplified 33.2

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

   15. Simplified 33.2

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

   17. Applied div-inv 33.2

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

   18. Applied add-sqr-sqrt 33.2

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

   19. Applied sqrt-prod 33.3

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

   20. Applied times-frac 33.2

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

   21. Simplified 33.1

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

   22. Taylor expanded around 0 33.5

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

   if 8.027827998243755e+98 < re

   1. Initial program 51.6

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

   3. Applied pow1 51.6

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

   4. Applied sqrt-pow1 51.6

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

   5. Applied log-pow 51.6

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

   6. Applied associate-/l* 51.6

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

   8. Applied pow1 51.6

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

   9. Applied log-pow 51.6

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

   10. Applied add-sqr-sqrt 51.6

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

   11. Applied times-frac 51.7

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

   12. Applied add-sqr-sqrt 51.7

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

   13. Applied times-frac 51.6

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

   14. Simplified 51.6

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

   15. Simplified 51.6

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

   16. Taylor expanded around inf 8.0

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

   17. Simplified 8.0

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\color{blue}{2 \cdot \log re}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.1975508038006968 \cdot 10^{153}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le -4.5405452937718227 \cdot 10^{-275}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \le 1.27662858127337166 \cdot 10^{-281}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt{\sqrt{\frac{1}{2}}}}{\sqrt{\log 10}} \cdot \left(\sqrt{\sqrt{\frac{1}{2}}} \cdot \left(2 \cdot \log im\right)\right)\right)\\ \mathbf{elif}\;re \le 8.0278279982437549 \cdot 10^{98}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{2 \cdot \log re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))