math.log/1 on complex, real part

Average Error: 31.2 → 17.5

Time: 1.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.154655280967186248169160366051090225892 \cdot 10^{151}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.0915017611570048756110314980016464877 \cdot 10^{-194}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 7.821508384246737745916295832197591042579 \cdot 10^{-215}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.893600070185494852606456234829585406642 \cdot 10^{120}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r88621 = re;
        double r88622 = r88621 * r88621;
        double r88623 = im;
        double r88624 = r88623 * r88623;
        double r88625 = r88622 + r88624;
        double r88626 = sqrt(r88625);
        double r88627 = log(r88626);
        return r88627;
}

↓

double f(double re, double im) {
        double r88628 = re;
        double r88629 = -4.154655280967186e+151;
        bool r88630 = r88628 <= r88629;
        double r88631 = -1.0;
        double r88632 = r88631 * r88628;
        double r88633 = log(r88632);
        double r88634 = -2.091501761157005e-194;
        bool r88635 = r88628 <= r88634;
        double r88636 = r88628 * r88628;
        double r88637 = im;
        double r88638 = r88637 * r88637;
        double r88639 = r88636 + r88638;
        double r88640 = sqrt(r88639);
        double r88641 = log(r88640);
        double r88642 = 7.821508384246738e-215;
        bool r88643 = r88628 <= r88642;
        double r88644 = log(r88637);
        double r88645 = 1.8936000701854949e+120;
        bool r88646 = r88628 <= r88645;
        double r88647 = log(r88628);
        double r88648 = r88646 ? r88641 : r88647;
        double r88649 = r88643 ? r88644 : r88648;
        double r88650 = r88635 ? r88641 : r88649;
        double r88651 = r88630 ? r88633 : r88650;
        return r88651;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -4.154655280967186e+151

   1. Initial program 63.5

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

   2. Taylor expanded around -inf 6.5

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

   if -4.154655280967186e+151 < re < -2.091501761157005e-194 or 7.821508384246738e-215 < re < 1.8936000701854949e+120

   1. Initial program 17.7

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

   if -2.091501761157005e-194 < re < 7.821508384246738e-215

   1. Initial program 31.1

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

   2. Taylor expanded around 0 33.5

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

   if 1.8936000701854949e+120 < re

   1. Initial program 56.2

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

   2. Taylor expanded around inf 7.7

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

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.154655280967186248169160366051090225892 \cdot 10^{151}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.0915017611570048756110314980016464877 \cdot 10^{-194}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 7.821508384246737745916295832197591042579 \cdot 10^{-215}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.893600070185494852606456234829585406642 \cdot 10^{120}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))