math.log/2 on complex, real part

Average Error: 32.0 → 17.8

Time: 25.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -2.263962099580682894408810458484502827138 \cdot 10^{113}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{im}\right)}{\log base}\\ \mathbf{elif}\;im \le 7.983252801042693681446527510289719577487 \cdot 10^{137}:\\ \;\;\;\;\frac{\left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left({\left(\log base\right)}^{2} + 0.0 \cdot 0.0\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{-\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r103095 = re;
        double r103096 = r103095 * r103095;
        double r103097 = im;
        double r103098 = r103097 * r103097;
        double r103099 = r103096 + r103098;
        double r103100 = sqrt(r103099);
        double r103101 = log(r103100);
        double r103102 = base;
        double r103103 = log(r103102);
        double r103104 = r103101 * r103103;
        double r103105 = atan2(r103097, r103095);
        double r103106 = 0.0;
        double r103107 = r103105 * r103106;
        double r103108 = r103104 + r103107;
        double r103109 = r103103 * r103103;
        double r103110 = r103106 * r103106;
        double r103111 = r103109 + r103110;
        double r103112 = r103108 / r103111;
        return r103112;
}

↓

double f(double re, double im, double base) {
        double r103113 = im;
        double r103114 = -2.263962099580683e+113;
        bool r103115 = r103113 <= r103114;
        double r103116 = -1.0;
        double r103117 = r103116 / r103113;
        double r103118 = log(r103117);
        double r103119 = -r103118;
        double r103120 = base;
        double r103121 = log(r103120);
        double r103122 = r103119 / r103121;
        double r103123 = 7.983252801042694e+137;
        bool r103124 = r103113 <= r103123;
        double r103125 = re;
        double r103126 = r103125 * r103125;
        double r103127 = r103113 * r103113;
        double r103128 = r103126 + r103127;
        double r103129 = sqrt(r103128);
        double r103130 = log(r103129);
        double r103131 = r103121 * r103130;
        double r103132 = r103131 * r103131;
        double r103133 = atan2(r103113, r103125);
        double r103134 = 0.0;
        double r103135 = r103133 * r103134;
        double r103136 = r103135 * r103135;
        double r103137 = r103132 - r103136;
        double r103138 = 2.0;
        double r103139 = pow(r103121, r103138);
        double r103140 = r103134 * r103134;
        double r103141 = r103139 + r103140;
        double r103142 = r103131 - r103135;
        double r103143 = r103141 * r103142;
        double r103144 = r103137 / r103143;
        double r103145 = log(r103113);
        double r103146 = -r103145;
        double r103147 = -r103121;
        double r103148 = r103146 / r103147;
        double r103149 = r103124 ? r103144 : r103148;
        double r103150 = r103115 ? r103122 : r103149;
        return r103150;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 3 regimes

2. if im < -2.263962099580683e+113

   1. Initial program 53.9

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

   3. Applied *-un-lft-identity 53.9

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

   4. Applied log-prod 53.9

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

   5. Applied distribute-lft-in 53.9

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

   6. Simplified 53.9

      \[\leadsto \frac{\left(\color{blue}{0} + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

   7. Simplified 53.9

      \[\leadsto \frac{\left(0 + \color{blue}{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 53.9

      \[\leadsto \frac{\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   10. Applied *-un-lft-identity 53.9

       \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   11. Applied times-frac 53.9

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   12. Simplified 53.9

       \[\leadsto \color{blue}{\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}} \cdot \frac{\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   13. Simplified 53.9

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

   14. Taylor expanded around -inf 64.0

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

   15. Simplified 7.6

       \[\leadsto \color{blue}{\frac{-\log \left(\frac{-1}{im}\right)}{0 + \log base}}\]

   if -2.263962099580683e+113 < im < 7.983252801042694e+137

   1. Initial program 22.0

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

   3. Applied *-un-lft-identity 22.0

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

   4. Applied log-prod 22.0

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

   5. Applied distribute-lft-in 22.0

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

   6. Simplified 22.0

      \[\leadsto \frac{\left(\color{blue}{0} + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

   7. Simplified 22.0

      \[\leadsto \frac{\left(0 + \color{blue}{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   8. Using strategy rm

   9. Applied flip-+ 22.0

      \[\leadsto \frac{\color{blue}{\frac{\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0}}}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

   10. Applied associate-/l/ 22.1

       \[\leadsto \color{blue}{\frac{\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(\log base \cdot \log base + 0.0 \cdot 0.0\right) \cdot \left(\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}\]

   11. Simplified 22.1

       \[\leadsto \frac{\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\color{blue}{\left({\left(\log base\right)}^{2} + 0.0 \cdot 0.0\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}\]

   if 7.983252801042694e+137 < im

   1. Initial program 59.1

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

   3. Applied *-un-lft-identity 59.1

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

   4. Applied log-prod 59.1

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

   5. Applied distribute-lft-in 59.1

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

   6. Simplified 59.1

      \[\leadsto \frac{\left(\color{blue}{0} + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

   7. Simplified 59.1

      \[\leadsto \frac{\left(0 + \color{blue}{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 59.1

      \[\leadsto \frac{\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   10. Applied *-un-lft-identity 59.1

       \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   11. Applied times-frac 59.1

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   12. Simplified 59.1

       \[\leadsto \color{blue}{\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}} \cdot \frac{\left(0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   13. Simplified 59.1

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

   14. Taylor expanded around inf 7.3

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

   15. Simplified 7.3

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -2.263962099580682894408810458484502827138 \cdot 10^{113}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{im}\right)}{\log base}\\ \mathbf{elif}\;im \le 7.983252801042693681446527510289719577487 \cdot 10^{137}:\\ \;\;\;\;\frac{\left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left({\left(\log base\right)}^{2} + 0.0 \cdot 0.0\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{-\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))