math.log/1 on complex, real part

Average Error: 30.8 → 17.3

Time: 3.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.3326130000804665 \cdot 10^{+121}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -3.263660527236801 \cdot 10^{-218}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 2.0545189035347466 \cdot 10^{-216}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.584078545341751 \cdot 10^{+99}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r579458 = re;
        double r579459 = r579458 * r579458;
        double r579460 = im;
        double r579461 = r579460 * r579460;
        double r579462 = r579459 + r579461;
        double r579463 = sqrt(r579462);
        double r579464 = log(r579463);
        return r579464;
}

↓

double f(double re, double im) {
        double r579465 = re;
        double r579466 = -5.3326130000804665e+121;
        bool r579467 = r579465 <= r579466;
        double r579468 = -r579465;
        double r579469 = log(r579468);
        double r579470 = -3.263660527236801e-218;
        bool r579471 = r579465 <= r579470;
        double r579472 = im;
        double r579473 = r579472 * r579472;
        double r579474 = r579465 * r579465;
        double r579475 = r579473 + r579474;
        double r579476 = sqrt(r579475);
        double r579477 = log(r579476);
        double r579478 = 2.0545189035347466e-216;
        bool r579479 = r579465 <= r579478;
        double r579480 = log(r579472);
        double r579481 = 2.584078545341751e+99;
        bool r579482 = r579465 <= r579481;
        double r579483 = log(r579465);
        double r579484 = r579482 ? r579477 : r579483;
        double r579485 = r579479 ? r579480 : r579484;
        double r579486 = r579471 ? r579477 : r579485;
        double r579487 = r579467 ? r579469 : r579486;
        return r579487;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -5.3326130000804665e+121

   1. Initial program 54.0

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

   2. Taylor expanded around -inf 7.7

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

   3. Simplified 7.7

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

   if -5.3326130000804665e+121 < re < -3.263660527236801e-218 or 2.0545189035347466e-216 < re < 2.584078545341751e+99

   1. Initial program 18.3

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

   if -3.263660527236801e-218 < re < 2.0545189035347466e-216

   1. Initial program 30.2

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

   2. Taylor expanded around 0 33.2

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

   if 2.584078545341751e+99 < re

   1. Initial program 50.4

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

   2. Taylor expanded around inf 8.8

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

4. Final simplification 17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.3326130000804665 \cdot 10^{+121}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -3.263660527236801 \cdot 10^{-218}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 2.0545189035347466 \cdot 10^{-216}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.584078545341751 \cdot 10^{+99}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))