math.log/1 on complex, real part

Average Error: 31.9 → 18.1

Time: 2.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -9.0150881706299043 \cdot 10^{107}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.28868009298563533 \cdot 10^{-244}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.3295675416621997 \cdot 10^{-138}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.6732898487300134 \cdot 10^{92}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r51411 = re;
        double r51412 = r51411 * r51411;
        double r51413 = im;
        double r51414 = r51413 * r51413;
        double r51415 = r51412 + r51414;
        double r51416 = sqrt(r51415);
        double r51417 = log(r51416);
        return r51417;
}

↓

double f(double re, double im) {
        double r51418 = re;
        double r51419 = -9.015088170629904e+107;
        bool r51420 = r51418 <= r51419;
        double r51421 = -r51418;
        double r51422 = log(r51421);
        double r51423 = -2.2886800929856353e-244;
        bool r51424 = r51418 <= r51423;
        double r51425 = r51418 * r51418;
        double r51426 = im;
        double r51427 = r51426 * r51426;
        double r51428 = r51425 + r51427;
        double r51429 = sqrt(r51428);
        double r51430 = log(r51429);
        double r51431 = 3.3295675416622e-138;
        bool r51432 = r51418 <= r51431;
        double r51433 = log(r51426);
        double r51434 = 5.673289848730013e+92;
        bool r51435 = r51418 <= r51434;
        double r51436 = log(r51418);
        double r51437 = r51435 ? r51430 : r51436;
        double r51438 = r51432 ? r51433 : r51437;
        double r51439 = r51424 ? r51430 : r51438;
        double r51440 = r51420 ? r51422 : r51439;
        return r51440;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -9.015088170629904e+107

   1. Initial program 52.7

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

   2. Taylor expanded around -inf 9.0

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

   3. Simplified 9.0

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

   if -9.015088170629904e+107 < re < -2.2886800929856353e-244 or 3.3295675416622e-138 < re < 5.673289848730013e+92

   1. Initial program 18.3

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

   if -2.2886800929856353e-244 < re < 3.3295675416622e-138

   1. Initial program 30.4

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

   2. Taylor expanded around 0 34.6

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

   if 5.673289848730013e+92 < re

   1. Initial program 49.9

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

   2. Taylor expanded around inf 8.7

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

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.0150881706299043 \cdot 10^{107}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.28868009298563533 \cdot 10^{-244}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.3295675416621997 \cdot 10^{-138}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.6732898487300134 \cdot 10^{92}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))