math.log/1 on complex, real part

Average Error: 32.2 → 17.9

Time: 3.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.199115232773121436978557182122313209889 \cdot 10^{121}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -9.538446928549385159760190961856882540607 \cdot 10^{-200}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.446990464932721506135775225301175117932 \cdot 10^{-197}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.107581686765815766066495864460425166206 \cdot 10^{61}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r23714 = re;
        double r23715 = r23714 * r23714;
        double r23716 = im;
        double r23717 = r23716 * r23716;
        double r23718 = r23715 + r23717;
        double r23719 = sqrt(r23718);
        double r23720 = log(r23719);
        return r23720;
}

↓

double f(double re, double im) {
        double r23721 = re;
        double r23722 = -1.1991152327731214e+121;
        bool r23723 = r23721 <= r23722;
        double r23724 = -1.0;
        double r23725 = r23724 * r23721;
        double r23726 = log(r23725);
        double r23727 = -9.538446928549385e-200;
        bool r23728 = r23721 <= r23727;
        double r23729 = r23721 * r23721;
        double r23730 = im;
        double r23731 = r23730 * r23730;
        double r23732 = r23729 + r23731;
        double r23733 = sqrt(r23732);
        double r23734 = log(r23733);
        double r23735 = 2.4469904649327215e-197;
        bool r23736 = r23721 <= r23735;
        double r23737 = log(r23730);
        double r23738 = 1.1075816867658158e+61;
        bool r23739 = r23721 <= r23738;
        double r23740 = log(r23721);
        double r23741 = r23739 ? r23734 : r23740;
        double r23742 = r23736 ? r23737 : r23741;
        double r23743 = r23728 ? r23734 : r23742;
        double r23744 = r23723 ? r23726 : r23743;
        return r23744;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.1991152327731214e+121

   1. Initial program 55.2

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

   2. Taylor expanded around -inf 7.5

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

   if -1.1991152327731214e+121 < re < -9.538446928549385e-200 or 2.4469904649327215e-197 < re < 1.1075816867658158e+61

   1. Initial program 18.5

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

   if -9.538446928549385e-200 < re < 2.4469904649327215e-197

   1. Initial program 31.8

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

   2. Taylor expanded around 0 33.7

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

   if 1.1075816867658158e+61 < re

   1. Initial program 46.9

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

   2. Taylor expanded around inf 10.6

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

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.199115232773121436978557182122313209889 \cdot 10^{121}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -9.538446928549385159760190961856882540607 \cdot 10^{-200}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.446990464932721506135775225301175117932 \cdot 10^{-197}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.107581686765815766066495864460425166206 \cdot 10^{61}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))