math.log/1 on complex, real part

Average Error: 31.2 → 17.4

Time: 2.5s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.8015950926867568 \cdot 10^{144}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.603232334857776 \cdot 10^{-212}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -5.74125170767144492 \cdot 10^{-228}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 4.4853367152010175 \cdot 10^{105}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r26288 = re;
        double r26289 = r26288 * r26288;
        double r26290 = im;
        double r26291 = r26290 * r26290;
        double r26292 = r26289 + r26291;
        double r26293 = sqrt(r26292);
        double r26294 = log(r26293);
        return r26294;
}

↓

double f(double re, double im) {
        double r26295 = re;
        double r26296 = -2.8015950926867568e+144;
        bool r26297 = r26295 <= r26296;
        double r26298 = -r26295;
        double r26299 = log(r26298);
        double r26300 = -2.6032323348577763e-212;
        bool r26301 = r26295 <= r26300;
        double r26302 = r26295 * r26295;
        double r26303 = im;
        double r26304 = r26303 * r26303;
        double r26305 = r26302 + r26304;
        double r26306 = sqrt(r26305);
        double r26307 = log(r26306);
        double r26308 = -5.741251707671445e-228;
        bool r26309 = r26295 <= r26308;
        double r26310 = 4.4853367152010175e+105;
        bool r26311 = r26295 <= r26310;
        double r26312 = log(r26295);
        double r26313 = r26311 ? r26307 : r26312;
        double r26314 = r26309 ? r26299 : r26313;
        double r26315 = r26301 ? r26307 : r26314;
        double r26316 = r26297 ? r26299 : r26315;
        return r26316;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -2.8015950926867568e+144 or -2.6032323348577763e-212 < re < -5.741251707671445e-228

   1. Initial program 57.8

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

   2. Taylor expanded around -inf 11.0

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

   3. Simplified 11.0

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

   if -2.8015950926867568e+144 < re < -2.6032323348577763e-212 or -5.741251707671445e-228 < re < 4.4853367152010175e+105

   1. Initial program 20.9

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

   if 4.4853367152010175e+105 < re

   1. Initial program 51.2

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

   2. Taylor expanded around inf 7.9

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

4. Final simplification 17.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.8015950926867568 \cdot 10^{144}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.603232334857776 \cdot 10^{-212}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -5.74125170767144492 \cdot 10^{-228}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 4.4853367152010175 \cdot 10^{105}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))