math.log/1 on complex, real part

Average Error: 32.5 → 17.8

Time: 2.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.1975508038006968 \cdot 10^{153}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -9.52817244882649108 \cdot 10^{-265}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.04745553524127593 \cdot 10^{-281}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.0421619395688348 \cdot 10^{92}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r37141 = re;
        double r37142 = r37141 * r37141;
        double r37143 = im;
        double r37144 = r37143 * r37143;
        double r37145 = r37142 + r37144;
        double r37146 = sqrt(r37145);
        double r37147 = log(r37146);
        return r37147;
}

↓

double f(double re, double im) {
        double r37148 = re;
        double r37149 = -4.197550803800697e+153;
        bool r37150 = r37148 <= r37149;
        double r37151 = -r37148;
        double r37152 = log(r37151);
        double r37153 = -9.528172448826491e-265;
        bool r37154 = r37148 <= r37153;
        double r37155 = r37148 * r37148;
        double r37156 = im;
        double r37157 = r37156 * r37156;
        double r37158 = r37155 + r37157;
        double r37159 = sqrt(r37158);
        double r37160 = log(r37159);
        double r37161 = 1.047455535241276e-281;
        bool r37162 = r37148 <= r37161;
        double r37163 = log(r37156);
        double r37164 = 8.042161939568835e+92;
        bool r37165 = r37148 <= r37164;
        double r37166 = log(r37148);
        double r37167 = r37165 ? r37160 : r37166;
        double r37168 = r37162 ? r37163 : r37167;
        double r37169 = r37154 ? r37160 : r37168;
        double r37170 = r37150 ? r37152 : r37169;
        return r37170;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -4.197550803800697e+153

   1. Initial program 63.9

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

   2. Taylor expanded around -inf 6.5

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

   3. Simplified 6.5

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

   if -4.197550803800697e+153 < re < -9.528172448826491e-265 or 1.047455535241276e-281 < re < 8.042161939568835e+92

   1. Initial program 21.3

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

   if -9.528172448826491e-265 < re < 1.047455535241276e-281

   1. Initial program 31.7

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

   2. Taylor expanded around 0 33.5

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

   if 8.042161939568835e+92 < re

   1. Initial program 50.4

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

   2. Taylor expanded around inf 7.8

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

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.1975508038006968 \cdot 10^{153}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -9.52817244882649108 \cdot 10^{-265}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.04745553524127593 \cdot 10^{-281}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.0421619395688348 \cdot 10^{92}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))