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 r36524 = re;
        double r36525 = r36524 * r36524;
        double r36526 = im;
        double r36527 = r36526 * r36526;
        double r36528 = r36525 + r36527;
        double r36529 = sqrt(r36528);
        double r36530 = log(r36529);
        return r36530;
}

↓

double f(double re, double im) {
        double r36531 = re;
        double r36532 = -2.8015950926867568e+144;
        bool r36533 = r36531 <= r36532;
        double r36534 = -r36531;
        double r36535 = log(r36534);
        double r36536 = -2.6032323348577763e-212;
        bool r36537 = r36531 <= r36536;
        double r36538 = r36531 * r36531;
        double r36539 = im;
        double r36540 = r36539 * r36539;
        double r36541 = r36538 + r36540;
        double r36542 = sqrt(r36541);
        double r36543 = log(r36542);
        double r36544 = -5.741251707671445e-228;
        bool r36545 = r36531 <= r36544;
        double r36546 = 4.4853367152010175e+105;
        bool r36547 = r36531 <= r36546;
        double r36548 = log(r36531);
        double r36549 = r36547 ? r36543 : r36548;
        double r36550 = r36545 ? r36535 : r36549;
        double r36551 = r36537 ? r36543 : r36550;
        double r36552 = r36533 ? r36535 : r36551;
        return r36552;
}

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)))))