math.log/1 on complex, real part

Average Error: 32.4 → 18.0

Time: 2.0s

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 r33072 = re;
        double r33073 = r33072 * r33072;
        double r33074 = im;
        double r33075 = r33074 * r33074;
        double r33076 = r33073 + r33075;
        double r33077 = sqrt(r33076);
        double r33078 = log(r33077);
        return r33078;
}

↓

double f(double re, double im) {
        double r33079 = re;
        double r33080 = -4.7575996220618e+138;
        bool r33081 = r33079 <= r33080;
        double r33082 = -r33079;
        double r33083 = log(r33082);
        double r33084 = -3.5543765182763856e-161;
        bool r33085 = r33079 <= r33084;
        double r33086 = r33079 * r33079;
        double r33087 = im;
        double r33088 = r33087 * r33087;
        double r33089 = r33086 + r33088;
        double r33090 = sqrt(r33089);
        double r33091 = log(r33090);
        double r33092 = 4.560703911778564e-251;
        bool r33093 = r33079 <= r33092;
        double r33094 = log(r33087);
        double r33095 = 3.266366167836414e+95;
        bool r33096 = r33079 <= r33095;
        double r33097 = log(r33079);
        double r33098 = r33096 ? r33091 : r33097;
        double r33099 = r33093 ? r33094 : r33098;
        double r33100 = r33085 ? r33091 : r33099;
        double r33101 = r33081 ? r33083 : r33100;
        return r33101;
}

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)))))