math.log10 on complex, real part

Average Error: 31.6 → 17.5

Time: 19.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.2033296293392884 \cdot 10^{+64}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{elif}\;re \le -1.2465139226076545 \cdot 10^{-304}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right)\\ \mathbf{elif}\;re \le 9.161820666739078 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{1}{2} \cdot \frac{\log 10}{\log im}}\\ \mathbf{elif}\;re \le 7.1259985451937965 \cdot 10^{+137}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log re \cdot 2\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r954943 = re;
        double r954944 = r954943 * r954943;
        double r954945 = im;
        double r954946 = r954945 * r954945;
        double r954947 = r954944 + r954946;
        double r954948 = sqrt(r954947);
        double r954949 = log(r954948);
        double r954950 = 10.0;
        double r954951 = log(r954950);
        double r954952 = r954949 / r954951;
        return r954952;
}

↓

double f(double re, double im) {
        double r954953 = re;
        double r954954 = -2.2033296293392884e+64;
        bool r954955 = r954953 <= r954954;
        double r954956 = 0.5;
        double r954957 = cbrt(r954956);
        double r954958 = 10.0;
        double r954959 = log(r954958);
        double r954960 = cbrt(r954959);
        double r954961 = r954957 / r954960;
        double r954962 = r954961 * r954961;
        double r954963 = -2.0;
        double r954964 = -1.0;
        double r954965 = r954964 / r954953;
        double r954966 = log(r954965);
        double r954967 = r954963 * r954966;
        double r954968 = r954967 * r954961;
        double r954969 = r954962 * r954968;
        double r954970 = -1.2465139226076545e-304;
        bool r954971 = r954953 <= r954970;
        double r954972 = im;
        double r954973 = r954972 * r954972;
        double r954974 = r954953 * r954953;
        double r954975 = r954973 + r954974;
        double r954976 = log(r954975);
        double r954977 = r954961 * r954976;
        double r954978 = r954962 * r954977;
        double r954979 = 9.161820666739078e-228;
        bool r954980 = r954953 <= r954979;
        double r954981 = log(r954972);
        double r954982 = r954959 / r954981;
        double r954983 = r954956 * r954982;
        double r954984 = r954956 / r954983;
        double r954985 = 7.1259985451937965e+137;
        bool r954986 = r954953 <= r954985;
        double r954987 = log(r954953);
        double r954988 = 2.0;
        double r954989 = r954987 * r954988;
        double r954990 = r954989 * r954961;
        double r954991 = r954990 * r954962;
        double r954992 = r954986 ? r954978 : r954991;
        double r954993 = r954980 ? r954984 : r954992;
        double r954994 = r954971 ? r954978 : r954993;
        double r954995 = r954955 ? r954969 : r954994;
        return r954995;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -2.2033296293392884e+64

   1. Initial program 45.5

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

   3. Applied pow1/2 45.5

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

   4. Applied log-pow 45.5

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

   5. Applied associate-/l* 45.5

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

   7. Applied pow1 45.5

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

   8. Applied log-pow 45.5

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

   9. Applied add-cube-cbrt 45.7

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

   10. Applied times-frac 45.8

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

   11. Applied add-cube-cbrt 45.5

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

   12. Applied times-frac 45.5

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

   13. Simplified 45.5

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

   14. Simplified 45.5

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

   15. Taylor expanded around -inf 10.6

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

   16. Simplified 10.6

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

   if -2.2033296293392884e+64 < re < -1.2465139226076545e-304 or 9.161820666739078e-228 < re < 7.1259985451937965e+137

   1. Initial program 20.2

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

   3. Applied pow1/2 20.2

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

   4. Applied log-pow 20.2

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

   5. Applied associate-/l* 20.3

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

   7. Applied pow1 20.3

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

   8. Applied log-pow 20.3

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

   9. Applied add-cube-cbrt 20.8

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

   10. Applied times-frac 20.8

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

   11. Applied add-cube-cbrt 20.2

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

   12. Applied times-frac 20.1

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

   13. Simplified 20.1

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

   14. Simplified 20.1

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

   if -1.2465139226076545e-304 < re < 9.161820666739078e-228

   1. Initial program 33.1

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

   3. Applied pow1/2 33.1

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

   4. Applied log-pow 33.1

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

   5. Applied associate-/l* 33.1

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

   6. Taylor expanded around 0 33.8

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{1}{2} \cdot \frac{\log 10}{\log im}}}\]

   if 7.1259985451937965e+137 < re

   1. Initial program 58.5

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

   3. Applied pow1/2 58.5

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

   4. Applied log-pow 58.5

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

   5. Applied associate-/l* 58.5

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

   7. Applied pow1 58.5

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

   8. Applied log-pow 58.5

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

   9. Applied add-cube-cbrt 58.5

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

   10. Applied times-frac 58.5

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

   11. Applied add-cube-cbrt 58.4

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

   12. Applied times-frac 58.4

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

   13. Simplified 58.4

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

   14. Simplified 58.4

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

   15. Taylor expanded around inf 8.1

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

   16. Simplified 8.1

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

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.2033296293392884 \cdot 10^{+64}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{elif}\;re \le -1.2465139226076545 \cdot 10^{-304}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right)\\ \mathbf{elif}\;re \le 9.161820666739078 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{1}{2} \cdot \frac{\log 10}{\log im}}\\ \mathbf{elif}\;re \le 7.1259985451937965 \cdot 10^{+137}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \log \left(im \cdot im + re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log re \cdot 2\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))