math.log/1 on complex, real part

Average Error: 31.5 → 18.1

Time: 8.8s

Precision: 64

Math

\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -8.915974928248364 \cdot 10^{+114}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.892315061161181 \cdot 10^{-257}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.475790091087918 \cdot 10^{-158}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 9.724903723406665 \cdot 10^{+105}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r1477165 = re;
        double r1477166 = r1477165 * r1477165;
        double r1477167 = im;
        double r1477168 = r1477167 * r1477167;
        double r1477169 = r1477166 + r1477168;
        double r1477170 = sqrt(r1477169);
        double r1477171 = log(r1477170);
        return r1477171;
}

↓

double f(double re, double im) {
        double r1477172 = re;
        double r1477173 = -8.915974928248364e+114;
        bool r1477174 = r1477172 <= r1477173;
        double r1477175 = -r1477172;
        double r1477176 = log(r1477175);
        double r1477177 = -4.892315061161181e-257;
        bool r1477178 = r1477172 <= r1477177;
        double r1477179 = im;
        double r1477180 = r1477179 * r1477179;
        double r1477181 = r1477172 * r1477172;
        double r1477182 = r1477180 + r1477181;
        double r1477183 = sqrt(r1477182);
        double r1477184 = log(r1477183);
        double r1477185 = 1.475790091087918e-158;
        bool r1477186 = r1477172 <= r1477185;
        double r1477187 = log(r1477179);
        double r1477188 = 9.724903723406665e+105;
        bool r1477189 = r1477172 <= r1477188;
        double r1477190 = log(r1477172);
        double r1477191 = r1477189 ? r1477184 : r1477190;
        double r1477192 = r1477186 ? r1477187 : r1477191;
        double r1477193 = r1477178 ? r1477184 : r1477192;
        double r1477194 = r1477174 ? r1477176 : r1477193;
        return r1477194;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -8.915974928248364e+114

   1. Initial program 53.9

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around -inf 9.3

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

   3. Simplified 9.3

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

   if -8.915974928248364e+114 < re < -4.892315061161181e-257 or 1.475790091087918e-158 < re < 9.724903723406665e+105

   1. Initial program 18.2

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   if -4.892315061161181e-257 < re < 1.475790091087918e-158

   1. Initial program 30.7

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around 0 35.3

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

   if 9.724903723406665e+105 < re

   1. Initial program 52.5

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around inf 8.9

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

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.915974928248364 \cdot 10^{+114}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.892315061161181 \cdot 10^{-257}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.475790091087918 \cdot 10^{-158}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 9.724903723406665 \cdot 10^{+105}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))