math.log/2 on complex, real part

Average Error: 32.3 → 18.8

Time: 9.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -0.04427952366159925573496636275194759946316:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -4.627946421707235249049212774387238708289 \cdot 10^{-181}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -2.244275880316184312571979912380930526646 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 149110596151586069466046352433323123408900:\\ \;\;\;\;\frac{1}{\sqrt{\sqrt[3]{{\left(\log base\right)}^{6}} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r53003 = re;
        double r53004 = r53003 * r53003;
        double r53005 = im;
        double r53006 = r53005 * r53005;
        double r53007 = r53004 + r53006;
        double r53008 = sqrt(r53007);
        double r53009 = log(r53008);
        double r53010 = base;
        double r53011 = log(r53010);
        double r53012 = r53009 * r53011;
        double r53013 = atan2(r53005, r53003);
        double r53014 = 0.0;
        double r53015 = r53013 * r53014;
        double r53016 = r53012 + r53015;
        double r53017 = r53011 * r53011;
        double r53018 = r53014 * r53014;
        double r53019 = r53017 + r53018;
        double r53020 = r53016 / r53019;
        return r53020;
}

↓

double f(double re, double im, double base) {
        double r53021 = re;
        double r53022 = -0.044279523661599256;
        bool r53023 = r53021 <= r53022;
        double r53024 = -1.0;
        double r53025 = r53024 * r53021;
        double r53026 = log(r53025);
        double r53027 = base;
        double r53028 = log(r53027);
        double r53029 = r53026 * r53028;
        double r53030 = im;
        double r53031 = atan2(r53030, r53021);
        double r53032 = 0.0;
        double r53033 = r53031 * r53032;
        double r53034 = r53029 + r53033;
        double r53035 = r53028 * r53028;
        double r53036 = r53032 * r53032;
        double r53037 = r53035 + r53036;
        double r53038 = r53034 / r53037;
        double r53039 = -4.627946421707235e-181;
        bool r53040 = r53021 <= r53039;
        double r53041 = r53021 * r53021;
        double r53042 = r53030 * r53030;
        double r53043 = r53041 + r53042;
        double r53044 = sqrt(r53043);
        double r53045 = log(r53044);
        double r53046 = r53045 * r53028;
        double r53047 = r53046 + r53033;
        double r53048 = 2.0;
        double r53049 = cbrt(r53027);
        double r53050 = log(r53049);
        double r53051 = r53048 * r53050;
        double r53052 = r53028 * r53051;
        double r53053 = r53028 * r53050;
        double r53054 = r53052 + r53053;
        double r53055 = r53054 + r53036;
        double r53056 = r53047 / r53055;
        double r53057 = -2.2442758803161843e-299;
        bool r53058 = r53021 <= r53057;
        double r53059 = 1.0;
        double r53060 = sqrt(r53037);
        double r53061 = r53059 / r53060;
        double r53062 = log(r53030);
        double r53063 = r53062 * r53028;
        double r53064 = r53063 + r53033;
        double r53065 = r53064 / r53060;
        double r53066 = r53061 * r53065;
        double r53067 = 1.4911059615158607e+41;
        bool r53068 = r53021 <= r53067;
        double r53069 = 6.0;
        double r53070 = pow(r53028, r53069);
        double r53071 = cbrt(r53070);
        double r53072 = r53071 + r53036;
        double r53073 = sqrt(r53072);
        double r53074 = r53059 / r53073;
        double r53075 = r53047 / r53060;
        double r53076 = r53074 * r53075;
        double r53077 = r53059 / r53021;
        double r53078 = log(r53077);
        double r53079 = r53059 / r53027;
        double r53080 = log(r53079);
        double r53081 = r53078 / r53080;
        double r53082 = r53068 ? r53076 : r53081;
        double r53083 = r53058 ? r53066 : r53082;
        double r53084 = r53040 ? r53056 : r53083;
        double r53085 = r53023 ? r53038 : r53084;
        return r53085;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if re < -0.044279523661599256

   1. Initial program 41.2

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

   2. Taylor expanded around -inf 13.5

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

   if -0.044279523661599256 < re < -4.627946421707235e-181

   1. Initial program 19.1

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

   3. Applied add-cube-cbrt 19.1

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

   4. Applied log-prod 19.2

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

   5. Applied distribute-lft-in 19.1

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

   6. Simplified 19.1

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

   if -4.627946421707235e-181 < re < -2.2442758803161843e-299

   1. Initial program 30.3

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

   3. Applied add-sqr-sqrt 30.3

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

   4. Applied *-un-lft-identity 30.3

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

   5. Applied times-frac 30.3

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

   6. Taylor expanded around 0 35.3

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

   if -2.2442758803161843e-299 < re < 1.4911059615158607e+41

   1. Initial program 23.1

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

   3. Applied add-sqr-sqrt 23.1

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

   4. Applied *-un-lft-identity 23.1

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

   5. Applied times-frac 23.1

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

   7. Applied add-cbrt-cube 23.2

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

   8. Applied add-cbrt-cube 23.2

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

   9. Applied cbrt-unprod 23.2

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

   10. Simplified 23.1

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

   if 1.4911059615158607e+41 < re

   1. Initial program 44.4

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

   2. Taylor expanded around inf 11.6

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

4. Final simplification 18.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -0.04427952366159925573496636275194759946316:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -4.627946421707235249049212774387238708289 \cdot 10^{-181}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -2.244275880316184312571979912380930526646 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 149110596151586069466046352433323123408900:\\ \;\;\;\;\frac{1}{\sqrt{\sqrt[3]{{\left(\log base\right)}^{6}} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(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))))