math.log/2 on complex, real part

Average Error: 31.1 → 16.9

Time: 44.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.144600521445903 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(-re\right)}}\\ \mathbf{elif}\;re \le -2.7249016889670465 \cdot 10^{-212}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{elif}\;re \le -9.044826970858667 \cdot 10^{-305}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 6.270425502185955 \cdot 10^{+140}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r1313110 = re;
        double r1313111 = r1313110 * r1313110;
        double r1313112 = im;
        double r1313113 = r1313112 * r1313112;
        double r1313114 = r1313111 + r1313113;
        double r1313115 = sqrt(r1313114);
        double r1313116 = log(r1313115);
        double r1313117 = base;
        double r1313118 = log(r1313117);
        double r1313119 = r1313116 * r1313118;
        double r1313120 = atan2(r1313112, r1313110);
        double r1313121 = 0.0;
        double r1313122 = r1313120 * r1313121;
        double r1313123 = r1313119 + r1313122;
        double r1313124 = r1313118 * r1313118;
        double r1313125 = r1313121 * r1313121;
        double r1313126 = r1313124 + r1313125;
        double r1313127 = r1313123 / r1313126;
        return r1313127;
}

↓

double f(double re, double im, double base) {
        double r1313128 = re;
        double r1313129 = -1.144600521445903e+98;
        bool r1313130 = r1313128 <= r1313129;
        double r1313131 = 1.0;
        double r1313132 = base;
        double r1313133 = log(r1313132);
        double r1313134 = -r1313128;
        double r1313135 = log(r1313134);
        double r1313136 = r1313133 / r1313135;
        double r1313137 = r1313131 / r1313136;
        double r1313138 = -2.7249016889670465e-212;
        bool r1313139 = r1313128 <= r1313138;
        double r1313140 = im;
        double r1313141 = r1313140 * r1313140;
        double r1313142 = r1313128 * r1313128;
        double r1313143 = r1313141 + r1313142;
        double r1313144 = sqrt(r1313143);
        double r1313145 = log(r1313144);
        double r1313146 = r1313133 / r1313145;
        double r1313147 = r1313131 / r1313146;
        double r1313148 = -9.044826970858667e-305;
        bool r1313149 = r1313128 <= r1313148;
        double r1313150 = log(r1313140);
        double r1313151 = r1313150 / r1313133;
        double r1313152 = 6.270425502185955e+140;
        bool r1313153 = r1313128 <= r1313152;
        double r1313154 = log(r1313128);
        double r1313155 = -r1313154;
        double r1313156 = -r1313133;
        double r1313157 = r1313155 / r1313156;
        double r1313158 = r1313153 ? r1313147 : r1313157;
        double r1313159 = r1313149 ? r1313151 : r1313158;
        double r1313160 = r1313139 ? r1313147 : r1313159;
        double r1313161 = r1313130 ? r1313137 : r1313160;
        return r1313161;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 4 regimes

2. if re < -1.144600521445903e+98

   1. Initial program 49.3

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

   2. Simplified 49.3

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

   4. Applied clear-num 49.3

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

   5. Simplified 49.2

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

   6. Taylor expanded around -inf 8.6

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

   7. Simplified 8.6

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

   if -1.144600521445903e+98 < re < -2.7249016889670465e-212 or -9.044826970858667e-305 < re < 6.270425502185955e+140

   1. Initial program 19.6

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

   2. Simplified 19.6

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

   4. Applied clear-num 19.6

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

   5. Simplified 19.6

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

   7. Applied add-cbrt-cube 19.7

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot \frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}\right) \cdot \frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}}\]
   8. Using strategy rm

   9. Applied frac-times 19.7

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

   10. Applied frac-times 19.7

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

   11. Applied cbrt-div 19.8

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

   12. Simplified 19.7

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

   13. Simplified 19.6

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

   if -2.7249016889670465e-212 < re < -9.044826970858667e-305

   1. Initial program 32.1

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

   2. Simplified 32.1

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

   4. Applied clear-num 32.1

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

   5. Simplified 32.0

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

   6. Taylor expanded around 0 32.7

      \[\leadsto \color{blue}{\frac{\log im}{\log base}}\]

   if 6.270425502185955e+140 < re

   1. Initial program 58.6

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

   2. Simplified 58.6

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

   4. Applied clear-num 58.6

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

   5. Simplified 58.6

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

   7. Applied div-inv 58.6

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

   8. Applied associate-/r* 58.6

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

   9. Taylor expanded around inf 6.5

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

   10. Simplified 6.5

       \[\leadsto \color{blue}{-\frac{\log re}{-\log base}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.144600521445903 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(-re\right)}}\\ \mathbf{elif}\;re \le -2.7249016889670465 \cdot 10^{-212}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{elif}\;re \le -9.044826970858667 \cdot 10^{-305}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 6.270425502185955 \cdot 10^{+140}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(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))))