math.log/2 on complex, real part

Average Error: 31.7 → 20.3

Time: 18.8s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -7.59904696746250469 \cdot 10^{146}:\\ \;\;\;\;\frac{1}{\sqrt{0.0 \cdot 0.0 + \sqrt[3]{{\left(\log base\right)}^{6}}}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(-re\right)}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\\ \mathbf{elif}\;re \le 5.36772892383815877 \cdot 10^{-207}:\\ \;\;\;\;\frac{1}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \left(\log base \cdot \log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right|\right) + \log base \cdot \log \left(\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)\right)}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\\ \mathbf{elif}\;re \le 2.1974199628881544 \cdot 10^{-67}:\\ \;\;\;\;\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log im}{{\left(\log base\right)}^{4} - {0.0}^{3} \cdot 0.0} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{elif}\;re \le 4130730593743547.5:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r76984 = re;
        double r76985 = r76984 * r76984;
        double r76986 = im;
        double r76987 = r76986 * r76986;
        double r76988 = r76985 + r76987;
        double r76989 = sqrt(r76988);
        double r76990 = log(r76989);
        double r76991 = base;
        double r76992 = log(r76991);
        double r76993 = r76990 * r76992;
        double r76994 = atan2(r76986, r76984);
        double r76995 = 0.0;
        double r76996 = r76994 * r76995;
        double r76997 = r76993 + r76996;
        double r76998 = r76992 * r76992;
        double r76999 = r76995 * r76995;
        double r77000 = r76998 + r76999;
        double r77001 = r76997 / r77000;
        return r77001;
}

↓

double f(double re, double im, double base) {
        double r77002 = re;
        double r77003 = -7.599046967462505e+146;
        bool r77004 = r77002 <= r77003;
        double r77005 = 1.0;
        double r77006 = 0.0;
        double r77007 = r77006 * r77006;
        double r77008 = base;
        double r77009 = log(r77008);
        double r77010 = 6.0;
        double r77011 = pow(r77009, r77010);
        double r77012 = cbrt(r77011);
        double r77013 = r77007 + r77012;
        double r77014 = sqrt(r77013);
        double r77015 = r77005 / r77014;
        double r77016 = im;
        double r77017 = atan2(r77016, r77002);
        double r77018 = r77017 * r77006;
        double r77019 = -r77002;
        double r77020 = log(r77019);
        double r77021 = r77009 * r77020;
        double r77022 = r77018 + r77021;
        double r77023 = 2.0;
        double r77024 = pow(r77009, r77023);
        double r77025 = r77007 + r77024;
        double r77026 = sqrt(r77025);
        double r77027 = r77022 / r77026;
        double r77028 = r77015 * r77027;
        double r77029 = 5.367728923838159e-207;
        bool r77030 = r77002 <= r77029;
        double r77031 = r77005 / r77026;
        double r77032 = r77002 * r77002;
        double r77033 = r77016 * r77016;
        double r77034 = r77032 + r77033;
        double r77035 = cbrt(r77034);
        double r77036 = fabs(r77035);
        double r77037 = log(r77036);
        double r77038 = r77009 * r77037;
        double r77039 = sqrt(r77035);
        double r77040 = log(r77039);
        double r77041 = r77009 * r77040;
        double r77042 = r77038 + r77041;
        double r77043 = r77018 + r77042;
        double r77044 = r77043 / r77026;
        double r77045 = r77031 * r77044;
        double r77046 = 2.1974199628881544e-67;
        bool r77047 = r77002 <= r77046;
        double r77048 = log(r77016);
        double r77049 = r77009 * r77048;
        double r77050 = r77018 + r77049;
        double r77051 = 4.0;
        double r77052 = pow(r77009, r77051);
        double r77053 = 3.0;
        double r77054 = pow(r77006, r77053);
        double r77055 = r77054 * r77006;
        double r77056 = r77052 - r77055;
        double r77057 = r77050 / r77056;
        double r77058 = r77009 * r77009;
        double r77059 = r77058 - r77007;
        double r77060 = r77057 * r77059;
        double r77061 = 4130730593743547.5;
        bool r77062 = r77002 <= r77061;
        double r77063 = sqrt(r77034);
        double r77064 = log(r77063);
        double r77065 = r77064 * r77009;
        double r77066 = r77065 + r77018;
        double r77067 = cbrt(r77008);
        double r77068 = log(r77067);
        double r77069 = r77023 * r77068;
        double r77070 = r77009 * r77069;
        double r77071 = r77009 * r77068;
        double r77072 = r77070 + r77071;
        double r77073 = r77072 + r77007;
        double r77074 = r77066 / r77073;
        double r77075 = log(r77002);
        double r77076 = -r77075;
        double r77077 = -r77009;
        double r77078 = r77076 / r77077;
        double r77079 = r77062 ? r77074 : r77078;
        double r77080 = r77047 ? r77060 : r77079;
        double r77081 = r77030 ? r77045 : r77080;
        double r77082 = r77004 ? r77028 : r77081;
        return r77082;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if re < -7.599046967462505e+146

   1. Initial program 62.5

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

   3. Applied add-sqr-sqrt 62.5

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

   4. Applied *-un-lft-identity 62.5

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

   5. Applied times-frac 62.5

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

   6. Simplified 62.5

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

   7. Simplified 62.5

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

   9. Applied add-cbrt-cube 62.5

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

   10. Simplified 62.5

       \[\leadsto \frac{1}{\sqrt{0.0 \cdot 0.0 + \sqrt[3]{\color{blue}{{\left(\log base\right)}^{6}}}}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\]

   11. Taylor expanded around -inf 7.5

       \[\leadsto \frac{1}{\sqrt{0.0 \cdot 0.0 + \sqrt[3]{{\left(\log base\right)}^{6}}}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \color{blue}{\left(-1 \cdot re\right)}}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\]

   12. Simplified 7.5

       \[\leadsto \frac{1}{\sqrt{0.0 \cdot 0.0 + \sqrt[3]{{\left(\log base\right)}^{6}}}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \color{blue}{\left(-re\right)}}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\]

   if -7.599046967462505e+146 < re < 5.367728923838159e-207

   1. Initial program 22.8

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

   3. Applied add-sqr-sqrt 22.8

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

   4. Applied *-un-lft-identity 22.8

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

   5. Applied times-frac 22.8

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

   6. Simplified 22.8

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

   7. Simplified 22.8

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

   9. Applied add-cube-cbrt 22.8

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

   10. Applied sqrt-prod 22.8

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

   11. Applied log-prod 22.9

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

   12. Applied distribute-lft-in 22.9

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

   13. Simplified 22.9

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

   if 5.367728923838159e-207 < re < 2.1974199628881544e-67

   1. Initial program 22.6

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

   3. Applied flip-+ 22.6

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

   4. Applied associate-/r/ 22.6

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

   5. Simplified 22.7

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{{\left(\log base\right)}^{4} - {0.0}^{3} \cdot 0.0}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\]

   6. Taylor expanded around 0 41.4

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \color{blue}{im}}{{\left(\log base\right)}^{4} - {0.0}^{3} \cdot 0.0} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\]

   if 2.1974199628881544e-67 < re < 4130730593743547.5

   1. Initial program 15.6

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

   3. Applied add-cube-cbrt 15.6

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

   4. Applied log-prod 15.6

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

   5. Applied distribute-lft-in 15.6

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

   6. Simplified 15.6

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

   if 4130730593743547.5 < re

   1. Initial program 41.1

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

   2. Taylor expanded around inf 13.3

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

   3. Simplified 13.3

      \[\leadsto \color{blue}{\frac{-\log re}{-\log base}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 20.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.59904696746250469 \cdot 10^{146}:\\ \;\;\;\;\frac{1}{\sqrt{0.0 \cdot 0.0 + \sqrt[3]{{\left(\log base\right)}^{6}}}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(-re\right)}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\\ \mathbf{elif}\;re \le 5.36772892383815877 \cdot 10^{-207}:\\ \;\;\;\;\frac{1}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \left(\log base \cdot \log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right|\right) + \log base \cdot \log \left(\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)\right)}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\\ \mathbf{elif}\;re \le 2.1974199628881544 \cdot 10^{-67}:\\ \;\;\;\;\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log im}{{\left(\log base\right)}^{4} - {0.0}^{3} \cdot 0.0} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{elif}\;re \le 4130730593743547.5:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))