math.log/1 on complex, real part

Average Error: 31.0 → 18.1

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.9126520588893617 \cdot 10^{+89}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.0485843961505591 \cdot 10^{-243}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.6776212883736178 \cdot 10^{-164}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3384984853102974.0:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r579525 = re;
        double r579526 = r579525 * r579525;
        double r579527 = im;
        double r579528 = r579527 * r579527;
        double r579529 = r579526 + r579528;
        double r579530 = sqrt(r579529);
        double r579531 = log(r579530);
        return r579531;
}

↓

double f(double re, double im) {
        double r579532 = re;
        double r579533 = -1.9126520588893617e+89;
        bool r579534 = r579532 <= r579533;
        double r579535 = -r579532;
        double r579536 = log(r579535);
        double r579537 = -1.0485843961505591e-243;
        bool r579538 = r579532 <= r579537;
        double r579539 = im;
        double r579540 = r579539 * r579539;
        double r579541 = r579532 * r579532;
        double r579542 = r579540 + r579541;
        double r579543 = sqrt(r579542);
        double r579544 = log(r579543);
        double r579545 = 1.6776212883736178e-164;
        bool r579546 = r579532 <= r579545;
        double r579547 = log(r579539);
        double r579548 = 3384984853102974.0;
        bool r579549 = r579532 <= r579548;
        double r579550 = log(r579532);
        double r579551 = r579549 ? r579544 : r579550;
        double r579552 = r579546 ? r579547 : r579551;
        double r579553 = r579538 ? r579544 : r579552;
        double r579554 = r579534 ? r579536 : r579553;
        return r579554;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.9126520588893617e+89

   1. Initial program 48.1

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

   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 -1.9126520588893617e+89 < re < -1.0485843961505591e-243 or 1.6776212883736178e-164 < re < 3384984853102974.0

   1. Initial program 18.5

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

   if -1.0485843961505591e-243 < re < 1.6776212883736178e-164

   1. Initial program 30.2

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

   2. Taylor expanded around 0 35.3

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

   if 3384984853102974.0 < re

   1. Initial program 40.1

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

   2. Taylor expanded around inf 11.6

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

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.9126520588893617 \cdot 10^{+89}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.0485843961505591 \cdot 10^{-243}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.6776212883736178 \cdot 10^{-164}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3384984853102974.0:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))