math.log10 on complex, real part

Average Error: 31.5 → 17.8

Time: 7.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.946647342182674517074717355805769171631 \cdot 10^{116}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -1.135823046566613171682980090169189733276 \cdot 10^{-194}:\\ \;\;\;\;\log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \frac{2}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\\ \mathbf{elif}\;re \le 7.357011145278623595340421361957804170614 \cdot 10^{-217}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 1.049250206408551427603735419097140899364 \cdot 10^{120}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left({\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}^{3}\right)}^{\frac{1}{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r52041 = re;
        double r52042 = r52041 * r52041;
        double r52043 = im;
        double r52044 = r52043 * r52043;
        double r52045 = r52042 + r52044;
        double r52046 = sqrt(r52045);
        double r52047 = log(r52046);
        double r52048 = 10.0;
        double r52049 = log(r52048);
        double r52050 = r52047 / r52049;
        return r52050;
}

↓

double f(double re, double im) {
        double r52051 = re;
        double r52052 = -3.9466473421826745e+116;
        bool r52053 = r52051 <= r52052;
        double r52054 = 1.0;
        double r52055 = 10.0;
        double r52056 = log(r52055);
        double r52057 = sqrt(r52056);
        double r52058 = r52054 / r52057;
        double r52059 = -1.0;
        double r52060 = r52059 / r52051;
        double r52061 = r52054 / r52056;
        double r52062 = sqrt(r52061);
        double r52063 = -r52062;
        double r52064 = pow(r52060, r52063);
        double r52065 = log(r52064);
        double r52066 = r52058 * r52065;
        double r52067 = -1.1358230465666132e-194;
        bool r52068 = r52051 <= r52067;
        double r52069 = r52051 * r52051;
        double r52070 = im;
        double r52071 = r52070 * r52070;
        double r52072 = r52069 + r52071;
        double r52073 = sqrt(r52072);
        double r52074 = pow(r52073, r52058);
        double r52075 = cbrt(r52074);
        double r52076 = log(r52075);
        double r52077 = 2.0;
        double r52078 = r52077 / r52057;
        double r52079 = r52076 * r52078;
        double r52080 = r52058 * r52076;
        double r52081 = r52079 + r52080;
        double r52082 = 7.357011145278624e-217;
        bool r52083 = r52051 <= r52082;
        double r52084 = pow(r52070, r52062);
        double r52085 = log(r52084);
        double r52086 = r52058 * r52085;
        double r52087 = 1.0492502064085514e+120;
        bool r52088 = r52051 <= r52087;
        double r52089 = 3.0;
        double r52090 = pow(r52074, r52089);
        double r52091 = 0.3333333333333333;
        double r52092 = pow(r52090, r52091);
        double r52093 = log(r52092);
        double r52094 = r52058 * r52093;
        double r52095 = r52054 / r52051;
        double r52096 = pow(r52095, r52063);
        double r52097 = log(r52096);
        double r52098 = r52058 * r52097;
        double r52099 = r52088 ? r52094 : r52098;
        double r52100 = r52083 ? r52086 : r52099;
        double r52101 = r52068 ? r52081 : r52100;
        double r52102 = r52053 ? r52066 : r52101;
        return r52102;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -3.9466473421826745e+116

   1. Initial program 53.2

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

   3. Applied add-sqr-sqrt 53.2

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

   4. Applied pow1 53.2

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

   5. Applied log-pow 53.2

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

   6. Applied times-frac 53.2

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

   8. Applied add-log-exp 53.2

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

   9. Simplified 53.1

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

   10. Taylor expanded around -inf 8.7

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

   11. Simplified 8.6

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

   if -3.9466473421826745e+116 < re < -1.1358230465666132e-194

   1. Initial program 17.4

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

   3. Applied add-sqr-sqrt 17.4

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

   4. Applied pow1 17.4

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

   5. Applied log-pow 17.4

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

   6. Applied times-frac 17.3

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

   8. Applied add-log-exp 17.3

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

   9. Simplified 17.1

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

   11. Applied add-cube-cbrt 17.2

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

   12. Applied log-prod 17.2

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

   13. Applied distribute-lft-in 17.2

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

   14. Simplified 17.2

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

   if -1.1358230465666132e-194 < re < 7.357011145278624e-217

   1. Initial program 31.3

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

   3. Applied add-sqr-sqrt 31.3

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

   4. Applied pow1 31.3

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

   5. Applied log-pow 31.3

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

   6. Applied times-frac 31.3

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

   8. Applied add-log-exp 31.3

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

   9. Simplified 31.2

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

   10. Taylor expanded around 0 33.8

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

   11. Simplified 33.7

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

   if 7.357011145278624e-217 < re < 1.0492502064085514e+120

   1. Initial program 19.2

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

   3. Applied add-sqr-sqrt 19.2

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

   4. Applied pow1 19.2

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

   5. Applied log-pow 19.2

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

   6. Applied times-frac 19.1

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

   8. Applied add-log-exp 19.1

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

   9. Simplified 19.0

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

   11. Applied add-cbrt-cube 19.0

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

   12. Simplified 19.0

       \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\color{blue}{{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}^{3}}}\right)\]
   13. Using strategy rm

   14. Applied pow1/3 19.1

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

   if 1.0492502064085514e+120 < re

   1. Initial program 56.1

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

   3. Applied add-sqr-sqrt 56.1

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

   4. Applied pow1 56.1

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

   5. Applied log-pow 56.1

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

   6. Applied times-frac 56.1

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

   8. Applied add-log-exp 56.1

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

   9. Simplified 56.1

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

   11. Applied add-cbrt-cube 56.1

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

   12. Simplified 56.1

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

   13. Taylor expanded around inf 8.0

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

   14. Simplified 7.9

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.946647342182674517074717355805769171631 \cdot 10^{116}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -1.135823046566613171682980090169189733276 \cdot 10^{-194}:\\ \;\;\;\;\log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \frac{2}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\\ \mathbf{elif}\;re \le 7.357011145278623595340421361957804170614 \cdot 10^{-217}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 1.049250206408551427603735419097140899364 \cdot 10^{120}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left({\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}^{3}\right)}^{\frac{1}{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))