math.log/1 on complex, real part

Average Error: 30.6 → 17.8

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.219316829266326 \cdot 10^{+151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -3.728827270067989 \cdot 10^{-149}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 2.0900313658610145 \cdot 10^{-225}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.944400793599736 \cdot 10^{+98}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r576975 = re;
        double r576976 = r576975 * r576975;
        double r576977 = im;
        double r576978 = r576977 * r576977;
        double r576979 = r576976 + r576978;
        double r576980 = sqrt(r576979);
        double r576981 = log(r576980);
        return r576981;
}

↓

double f(double re, double im) {
        double r576982 = re;
        double r576983 = -3.219316829266326e+151;
        bool r576984 = r576982 <= r576983;
        double r576985 = -r576982;
        double r576986 = log(r576985);
        double r576987 = -3.728827270067989e-149;
        bool r576988 = r576982 <= r576987;
        double r576989 = im;
        double r576990 = r576989 * r576989;
        double r576991 = r576982 * r576982;
        double r576992 = r576990 + r576991;
        double r576993 = sqrt(r576992);
        double r576994 = log(r576993);
        double r576995 = 2.0900313658610145e-225;
        bool r576996 = r576982 <= r576995;
        double r576997 = log(r576989);
        double r576998 = 6.944400793599736e+98;
        bool r576999 = r576982 <= r576998;
        double r577000 = log(r576982);
        double r577001 = r576999 ? r576994 : r577000;
        double r577002 = r576996 ? r576997 : r577001;
        double r577003 = r576988 ? r576994 : r577002;
        double r577004 = r576984 ? r576986 : r577003;
        return r577004;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -3.219316829266326e+151

   1. Initial program 61.3

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

   2. Taylor expanded around -inf 7.0

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

   3. Simplified 7.0

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

   if -3.219316829266326e+151 < re < -3.728827270067989e-149 or 2.0900313658610145e-225 < re < 6.944400793599736e+98

   1. Initial program 16.9

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

   if -3.728827270067989e-149 < re < 2.0900313658610145e-225

   1. Initial program 30.1

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

   2. Taylor expanded around 0 34.9

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

   if 6.944400793599736e+98 < re

   1. Initial program 49.8

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

   2. Taylor expanded around inf 8.6

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.219316829266326 \cdot 10^{+151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -3.728827270067989 \cdot 10^{-149}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 2.0900313658610145 \cdot 10^{-225}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.944400793599736 \cdot 10^{+98}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))