math.log/1 on complex, real part

Average Error: 30.5 → 17.7

Time: 2.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.2048133479194113 \cdot 10^{+151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -7.643262657340371 \cdot 10^{-261}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.5865420288945386 \cdot 10^{-158}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.327483106107868 \cdot 10^{+105}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r1151214 = re;
        double r1151215 = r1151214 * r1151214;
        double r1151216 = im;
        double r1151217 = r1151216 * r1151216;
        double r1151218 = r1151215 + r1151217;
        double r1151219 = sqrt(r1151218);
        double r1151220 = log(r1151219);
        return r1151220;
}

↓

double f(double re, double im) {
        double r1151221 = re;
        double r1151222 = -1.2048133479194113e+151;
        bool r1151223 = r1151221 <= r1151222;
        double r1151224 = -r1151221;
        double r1151225 = log(r1151224);
        double r1151226 = -7.643262657340371e-261;
        bool r1151227 = r1151221 <= r1151226;
        double r1151228 = im;
        double r1151229 = r1151228 * r1151228;
        double r1151230 = r1151221 * r1151221;
        double r1151231 = r1151229 + r1151230;
        double r1151232 = sqrt(r1151231);
        double r1151233 = log(r1151232);
        double r1151234 = 1.5865420288945386e-158;
        bool r1151235 = r1151221 <= r1151234;
        double r1151236 = log(r1151228);
        double r1151237 = 1.327483106107868e+105;
        bool r1151238 = r1151221 <= r1151237;
        double r1151239 = log(r1151221);
        double r1151240 = r1151238 ? r1151233 : r1151239;
        double r1151241 = r1151235 ? r1151236 : r1151240;
        double r1151242 = r1151227 ? r1151233 : r1151241;
        double r1151243 = r1151223 ? r1151225 : r1151242;
        return r1151243;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.2048133479194113e+151

   1. Initial program 61.2

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

   2. Taylor expanded around -inf 7.4

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

   3. Simplified 7.4

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

   if -1.2048133479194113e+151 < re < -7.643262657340371e-261 or 1.5865420288945386e-158 < re < 1.327483106107868e+105

   1. Initial program 17.6

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

   if -7.643262657340371e-261 < re < 1.5865420288945386e-158

   1. Initial program 29.8

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

   2. Taylor expanded around 0 34.7

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

   if 1.327483106107868e+105 < re

   1. Initial program 50.7

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

   2. Taylor expanded around inf 8.9

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

4. Final simplification 17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2048133479194113 \cdot 10^{+151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -7.643262657340371 \cdot 10^{-261}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.5865420288945386 \cdot 10^{-158}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.327483106107868 \cdot 10^{+105}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

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