math.log/1 on complex, real part

Average Error: 31.3 → 17.2

Time: 6.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.2033296293392884 \cdot 10^{+64}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.2465139226076545 \cdot 10^{-304}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 9.161820666739078 \cdot 10^{-228}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 7.1259985451937965 \cdot 10^{+137}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r1709611 = re;
        double r1709612 = r1709611 * r1709611;
        double r1709613 = im;
        double r1709614 = r1709613 * r1709613;
        double r1709615 = r1709612 + r1709614;
        double r1709616 = sqrt(r1709615);
        double r1709617 = log(r1709616);
        return r1709617;
}

↓

double f(double re, double im) {
        double r1709618 = re;
        double r1709619 = -2.2033296293392884e+64;
        bool r1709620 = r1709618 <= r1709619;
        double r1709621 = -r1709618;
        double r1709622 = log(r1709621);
        double r1709623 = -1.2465139226076545e-304;
        bool r1709624 = r1709618 <= r1709623;
        double r1709625 = im;
        double r1709626 = r1709625 * r1709625;
        double r1709627 = r1709618 * r1709618;
        double r1709628 = r1709626 + r1709627;
        double r1709629 = sqrt(r1709628);
        double r1709630 = log(r1709629);
        double r1709631 = 9.161820666739078e-228;
        bool r1709632 = r1709618 <= r1709631;
        double r1709633 = log(r1709625);
        double r1709634 = 7.1259985451937965e+137;
        bool r1709635 = r1709618 <= r1709634;
        double r1709636 = log(r1709618);
        double r1709637 = r1709635 ? r1709630 : r1709636;
        double r1709638 = r1709632 ? r1709633 : r1709637;
        double r1709639 = r1709624 ? r1709630 : r1709638;
        double r1709640 = r1709620 ? r1709622 : r1709639;
        return r1709640;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -2.2033296293392884e+64

   1. Initial program 45.4

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

   2. Taylor expanded around -inf 10.3

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

   3. Simplified 10.3

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

   if -2.2033296293392884e+64 < re < -1.2465139226076545e-304 or 9.161820666739078e-228 < re < 7.1259985451937965e+137

   1. Initial program 19.8

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

   if -1.2465139226076545e-304 < re < 9.161820666739078e-228

   1. Initial program 32.9

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

   2. Taylor expanded around 0 33.5

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

   if 7.1259985451937965e+137 < re

   1. Initial program 58.4

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

   2. Taylor expanded around inf 7.8

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

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.2033296293392884 \cdot 10^{+64}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.2465139226076545 \cdot 10^{-304}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 9.161820666739078 \cdot 10^{-228}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 7.1259985451937965 \cdot 10^{+137}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))