math.log/1 on complex, real part

Average Error: 30.7 → 17.1

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.7109531485520302 \cdot 10^{+90}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -5.7290891404837934 \cdot 10^{-198}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le -6.376038522303672 \cdot 10^{-268}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.7061266383920343 \cdot 10^{+78}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r3337201 = re;
        double r3337202 = r3337201 * r3337201;
        double r3337203 = im;
        double r3337204 = r3337203 * r3337203;
        double r3337205 = r3337202 + r3337204;
        double r3337206 = sqrt(r3337205);
        double r3337207 = log(r3337206);
        return r3337207;
}

↓

double f(double re, double im) {
        double r3337208 = re;
        double r3337209 = -1.7109531485520302e+90;
        bool r3337210 = r3337208 <= r3337209;
        double r3337211 = -r3337208;
        double r3337212 = log(r3337211);
        double r3337213 = -5.7290891404837934e-198;
        bool r3337214 = r3337208 <= r3337213;
        double r3337215 = im;
        double r3337216 = r3337215 * r3337215;
        double r3337217 = r3337208 * r3337208;
        double r3337218 = r3337216 + r3337217;
        double r3337219 = sqrt(r3337218);
        double r3337220 = log(r3337219);
        double r3337221 = -6.376038522303672e-268;
        bool r3337222 = r3337208 <= r3337221;
        double r3337223 = log(r3337215);
        double r3337224 = 1.7061266383920343e+78;
        bool r3337225 = r3337208 <= r3337224;
        double r3337226 = log(r3337208);
        double r3337227 = r3337225 ? r3337220 : r3337226;
        double r3337228 = r3337222 ? r3337223 : r3337227;
        double r3337229 = r3337214 ? r3337220 : r3337228;
        double r3337230 = r3337210 ? r3337212 : r3337229;
        return r3337230;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.7109531485520302e+90

   1. Initial program 48.3

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

   2. Taylor expanded around -inf 8.6

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

   3. Simplified 8.6

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

   if -1.7109531485520302e+90 < re < -5.7290891404837934e-198 or -6.376038522303672e-268 < re < 1.7061266383920343e+78

   1. Initial program 20.3

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

   if -5.7290891404837934e-198 < re < -6.376038522303672e-268

   1. Initial program 29.4

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

   2. Taylor expanded around 0 34.7

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

   if 1.7061266383920343e+78 < re

   1. Initial program 46.3

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

   2. Taylor expanded around inf 10.3

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.7109531485520302 \cdot 10^{+90}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -5.7290891404837934 \cdot 10^{-198}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le -6.376038522303672 \cdot 10^{-268}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.7061266383920343 \cdot 10^{+78}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))