math.log/2 on complex, real part

Average Error: 32.4 → 17.7

Time: 7.9s

Precision: 64

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -7.07557702361520036 \cdot 10^{54}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -6.3360595030704328 \cdot 10^{-287}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}\\ \mathbf{elif}\;re \le 1.1087233082144876 \cdot 10^{-259}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 4.4918944413865042 \cdot 10^{101}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r43963 = re;
        double r43964 = r43963 * r43963;
        double r43965 = im;
        double r43966 = r43965 * r43965;
        double r43967 = r43964 + r43966;
        double r43968 = sqrt(r43967);
        double r43969 = log(r43968);
        double r43970 = base;
        double r43971 = log(r43970);
        double r43972 = r43969 * r43971;
        double r43973 = atan2(r43965, r43963);
        double r43974 = 0.0;
        double r43975 = r43973 * r43974;
        double r43976 = r43972 + r43975;
        double r43977 = r43971 * r43971;
        double r43978 = r43974 * r43974;
        double r43979 = r43977 + r43978;
        double r43980 = r43976 / r43979;
        return r43980;
}

↓

double f(double re, double im, double base) {
        double r43981 = re;
        double r43982 = -7.0755770236152e+54;
        bool r43983 = r43981 <= r43982;
        double r43984 = -1.0;
        double r43985 = r43984 * r43981;
        double r43986 = log(r43985);
        double r43987 = base;
        double r43988 = log(r43987);
        double r43989 = r43986 * r43988;
        double r43990 = im;
        double r43991 = atan2(r43990, r43981);
        double r43992 = 0.0;
        double r43993 = r43991 * r43992;
        double r43994 = r43989 + r43993;
        double r43995 = r43988 * r43988;
        double r43996 = r43992 * r43992;
        double r43997 = r43995 + r43996;
        double r43998 = r43994 / r43997;
        double r43999 = -6.336059503070433e-287;
        bool r44000 = r43981 <= r43999;
        double r44001 = 1.0;
        double r44002 = cbrt(r44001);
        double r44003 = r44002 * r44002;
        double r44004 = sqrt(r44001);
        double r44005 = r44003 / r44004;
        double r44006 = r43981 * r43981;
        double r44007 = r43990 * r43990;
        double r44008 = r44006 + r44007;
        double r44009 = sqrt(r44008);
        double r44010 = log(r44009);
        double r44011 = r44010 * r43988;
        double r44012 = r44011 + r43993;
        double r44013 = 2.0;
        double r44014 = pow(r43988, r44013);
        double r44015 = r43996 + r44014;
        double r44016 = r44012 / r44015;
        double r44017 = r44005 * r44016;
        double r44018 = 1.1087233082144876e-259;
        bool r44019 = r43981 <= r44018;
        double r44020 = log(r43990);
        double r44021 = r44020 / r43988;
        double r44022 = r44005 * r44021;
        double r44023 = 4.491894441386504e+101;
        bool r44024 = r43981 <= r44023;
        double r44025 = r44001 / r43981;
        double r44026 = log(r44025);
        double r44027 = r44001 / r43987;
        double r44028 = log(r44027);
        double r44029 = r44026 / r44028;
        double r44030 = r44005 * r44029;
        double r44031 = r44024 ? r44017 : r44030;
        double r44032 = r44019 ? r44022 : r44031;
        double r44033 = r44000 ? r44017 : r44032;
        double r44034 = r43983 ? r43998 : r44033;
        return r44034;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 4 regimes

2. if re < -7.0755770236152e+54

   1. Initial program 46.2

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

   2. Taylor expanded around -inf 11.5

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

   if -7.0755770236152e+54 < re < -6.336059503070433e-287 or 1.1087233082144876e-259 < re < 4.491894441386504e+101

   1. Initial program 20.8

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 20.8

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   4. Applied *-un-lft-identity 20.8

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   5. Applied times-frac 20.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 20.8

      \[\leadsto \frac{1}{\sqrt{\color{blue}{1 \cdot \left(\log base \cdot \log base + 0.0 \cdot 0.0\right)}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   8. Applied sqrt-prod 20.8

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   9. Applied add-cube-cbrt 20.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{1} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   10. Applied times-frac 20.8

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\right)} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   11. Applied associate-*l* 20.8

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\right)}\]

   12. Simplified 20.8

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\]

   if -6.336059503070433e-287 < re < 1.1087233082144876e-259

   1. Initial program 32.6

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 32.6

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   4. Applied *-un-lft-identity 32.6

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   5. Applied times-frac 32.6

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 32.6

      \[\leadsto \frac{1}{\sqrt{\color{blue}{1 \cdot \left(\log base \cdot \log base + 0.0 \cdot 0.0\right)}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   8. Applied sqrt-prod 32.6

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   9. Applied add-cube-cbrt 32.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{1} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   10. Applied times-frac 32.6

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\right)} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   11. Applied associate-*l* 32.6

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\right)}\]

   12. Simplified 32.6

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\]

   13. Taylor expanded around 0 34.3

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \color{blue}{\frac{\log im}{\log base}}\]

   if 4.491894441386504e+101 < re

   1. Initial program 53.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 53.3

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   4. Applied *-un-lft-identity 53.3

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   5. Applied times-frac 53.3

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 53.3

      \[\leadsto \frac{1}{\sqrt{\color{blue}{1 \cdot \left(\log base \cdot \log base + 0.0 \cdot 0.0\right)}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   8. Applied sqrt-prod 53.3

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   9. Applied add-cube-cbrt 53.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{1} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   10. Applied times-frac 53.3

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\right)} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   11. Applied associate-*l* 53.3

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\right)}\]

   12. Simplified 53.3

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\]

   13. Taylor expanded around inf 9.1

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

4. Final simplification 17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.07557702361520036 \cdot 10^{54}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -6.3360595030704328 \cdot 10^{-287}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}\\ \mathbf{elif}\;re \le 1.1087233082144876 \cdot 10^{-259}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 4.4918944413865042 \cdot 10^{101}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1}} \cdot \frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))