math.log/2 on complex, real part

Average Error: 31.0 → 18.6

Time: 1.1m

Precision: 64

• could not determine a ground truth for program body

  1. re = 5.0915677267012865e-76
  2. im = -6.759561038453283e+75
  3. base = 1.2325639830328598e-68

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

↓

\[\begin{array}{l} \mathbf{if}\;im \le -8.551551215711193 \cdot 10^{+156}:\\ \;\;\;\;\left(-\log \left(\frac{-1}{re}\right)\right) \cdot \sqrt[3]{\frac{1}{\log base \cdot \left(\log base \cdot \log base\right)}}\\ \mathbf{elif}\;im \le -2.108728536359279 \cdot 10^{-15}:\\ \;\;\;\;\sqrt[3]{\left(\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base} \cdot \frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}\right) \cdot \frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\\ \mathbf{elif}\;im \le 2.4294881236333897 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;im \le 0.0192463786380111:\\ \;\;\;\;\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}{\log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \log base + \log base \cdot \log \left(\sqrt[3]{base}\right)}\\ \mathbf{elif}\;im \le 3.5464671902525163 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r1405142 = re;
        double r1405143 = r1405142 * r1405142;
        double r1405144 = im;
        double r1405145 = r1405144 * r1405144;
        double r1405146 = r1405143 + r1405145;
        double r1405147 = sqrt(r1405146);
        double r1405148 = log(r1405147);
        double r1405149 = base;
        double r1405150 = log(r1405149);
        double r1405151 = r1405148 * r1405150;
        double r1405152 = atan2(r1405144, r1405142);
        double r1405153 = 0.0;
        double r1405154 = r1405152 * r1405153;
        double r1405155 = r1405151 + r1405154;
        double r1405156 = r1405150 * r1405150;
        double r1405157 = r1405153 * r1405153;
        double r1405158 = r1405156 + r1405157;
        double r1405159 = r1405155 / r1405158;
        return r1405159;
}

↓

double f(double re, double im, double base) {
        double r1405160 = im;
        double r1405161 = -8.551551215711193e+156;
        bool r1405162 = r1405160 <= r1405161;
        double r1405163 = -1.0;
        double r1405164 = re;
        double r1405165 = r1405163 / r1405164;
        double r1405166 = log(r1405165);
        double r1405167 = -r1405166;
        double r1405168 = 1.0;
        double r1405169 = base;
        double r1405170 = log(r1405169);
        double r1405171 = r1405170 * r1405170;
        double r1405172 = r1405170 * r1405171;
        double r1405173 = r1405168 / r1405172;
        double r1405174 = cbrt(r1405173);
        double r1405175 = r1405167 * r1405174;
        double r1405176 = -2.108728536359279e-15;
        bool r1405177 = r1405160 <= r1405176;
        double r1405178 = r1405160 * r1405160;
        double r1405179 = r1405164 * r1405164;
        double r1405180 = r1405178 + r1405179;
        double r1405181 = sqrt(r1405180);
        double r1405182 = log(r1405181);
        double r1405183 = r1405182 / r1405170;
        double r1405184 = r1405183 * r1405183;
        double r1405185 = r1405184 * r1405183;
        double r1405186 = cbrt(r1405185);
        double r1405187 = 2.4294881236333897e-88;
        bool r1405188 = r1405160 <= r1405187;
        double r1405189 = r1405167 / r1405170;
        double r1405190 = 0.0192463786380111;
        bool r1405191 = r1405160 <= r1405190;
        double r1405192 = r1405182 * r1405170;
        double r1405193 = cbrt(r1405169);
        double r1405194 = r1405193 * r1405193;
        double r1405195 = log(r1405194);
        double r1405196 = r1405195 * r1405170;
        double r1405197 = log(r1405193);
        double r1405198 = r1405170 * r1405197;
        double r1405199 = r1405196 + r1405198;
        double r1405200 = r1405192 / r1405199;
        double r1405201 = 3.5464671902525163e+61;
        bool r1405202 = r1405160 <= r1405201;
        double r1405203 = log(r1405160);
        double r1405204 = r1405203 / r1405170;
        double r1405205 = r1405202 ? r1405189 : r1405204;
        double r1405206 = r1405191 ? r1405200 : r1405205;
        double r1405207 = r1405188 ? r1405189 : r1405206;
        double r1405208 = r1405177 ? r1405186 : r1405207;
        double r1405209 = r1405162 ? r1405175 : r1405208;
        return r1405209;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if im < -8.551551215711193e+156

   1. Initial program 62.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 62.0

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

   3. Taylor expanded around -inf 62.8

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

   4. Simplified 52.2

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

   5. Taylor expanded around -inf 52.2

      \[\leadsto -\frac{\color{blue}{\log \left(\frac{-1}{re}\right)}}{\log base}\]
   6. Using strategy rm

   7. Applied div-inv 52.2

      \[\leadsto -\color{blue}{\log \left(\frac{-1}{re}\right) \cdot \frac{1}{\log base}}\]
   8. Using strategy rm

   9. Applied add-cbrt-cube 52.2

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

   10. Applied add-cbrt-cube 52.2

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

   11. Applied cbrt-undiv 52.2

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

   12. Simplified 52.2

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

   if -8.551551215711193e+156 < im < -2.108728536359279e-15

   1. Initial program 18.2

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 18.2

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

   4. Applied add-cbrt-cube 18.4

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

   5. Applied add-cbrt-cube 18.6

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

   6. Applied cbrt-unprod 18.4

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

   7. Applied add-cbrt-cube 18.5

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

   8. Applied add-cbrt-cube 18.5

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

   9. Applied cbrt-unprod 18.4

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

   10. Applied cbrt-undiv 18.4

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

   11. Simplified 18.4

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

   if -2.108728536359279e-15 < im < 2.4294881236333897e-88 or 0.0192463786380111 < im < 3.5464671902525163e+61

   1. Initial program 23.2

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 23.2

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

   3. Taylor expanded around -inf 62.8

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

   4. Simplified 13.7

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

   5. Taylor expanded around -inf 13.7

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

   if 2.4294881236333897e-88 < im < 0.0192463786380111

   1. Initial program 17.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 17.0

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

   4. Applied add-cube-cbrt 17.0

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

   5. Applied log-prod 17.0

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

   6. Applied distribute-rgt-in 17.0

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

   if 3.5464671902525163e+61 < im

   1. Initial program 44.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 44.3

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

   3. Taylor expanded around 0 10.5

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

4. Final simplification 18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -8.551551215711193 \cdot 10^{+156}:\\ \;\;\;\;\left(-\log \left(\frac{-1}{re}\right)\right) \cdot \sqrt[3]{\frac{1}{\log base \cdot \left(\log base \cdot \log base\right)}}\\ \mathbf{elif}\;im \le -2.108728536359279 \cdot 10^{-15}:\\ \;\;\;\;\sqrt[3]{\left(\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base} \cdot \frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}\right) \cdot \frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\\ \mathbf{elif}\;im \le 2.4294881236333897 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;im \le 0.0192463786380111:\\ \;\;\;\;\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right) \cdot \log base}{\log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \log base + \log base \cdot \log \left(\sqrt[3]{base}\right)}\\ \mathbf{elif}\;im \le 3.5464671902525163 \cdot 10^{+61}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0))))