math.log/1 on complex, real part

Average Error: 32.4 → 18.0

Time: 2.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.75759962206180014 \cdot 10^{138}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -3.5543765182763856 \cdot 10^{-161}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.5607039117785637 \cdot 10^{-251}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.2663661678364143 \cdot 10^{95}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r35213 = re;
        double r35214 = r35213 * r35213;
        double r35215 = im;
        double r35216 = r35215 * r35215;
        double r35217 = r35214 + r35216;
        double r35218 = sqrt(r35217);
        double r35219 = log(r35218);
        return r35219;
}

↓

double f(double re, double im) {
        double r35220 = re;
        double r35221 = -4.7575996220618e+138;
        bool r35222 = r35220 <= r35221;
        double r35223 = -r35220;
        double r35224 = log(r35223);
        double r35225 = -3.5543765182763856e-161;
        bool r35226 = r35220 <= r35225;
        double r35227 = r35220 * r35220;
        double r35228 = im;
        double r35229 = r35228 * r35228;
        double r35230 = r35227 + r35229;
        double r35231 = sqrt(r35230);
        double r35232 = log(r35231);
        double r35233 = 4.560703911778564e-251;
        bool r35234 = r35220 <= r35233;
        double r35235 = log(r35228);
        double r35236 = 3.266366167836414e+95;
        bool r35237 = r35220 <= r35236;
        double r35238 = log(r35220);
        double r35239 = r35237 ? r35232 : r35238;
        double r35240 = r35234 ? r35235 : r35239;
        double r35241 = r35226 ? r35232 : r35240;
        double r35242 = r35222 ? r35224 : r35241;
        return r35242;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -4.7575996220618e+138

   1. Initial program 59.1

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

   2. Taylor expanded around -inf 7.1

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

   3. Simplified 7.1

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

   if -4.7575996220618e+138 < re < -3.5543765182763856e-161 or 4.560703911778564e-251 < re < 3.266366167836414e+95

   1. Initial program 19.0

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

   if -3.5543765182763856e-161 < re < 4.560703911778564e-251

   1. Initial program 32.9

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

   2. Taylor expanded around 0 33.8

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

   if 3.266366167836414e+95 < re

   1. Initial program 51.1

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

   2. Taylor expanded around inf 9.2

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.75759962206180014 \cdot 10^{138}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -3.5543765182763856 \cdot 10^{-161}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 4.5607039117785637 \cdot 10^{-251}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.2663661678364143 \cdot 10^{95}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))