math.log/1 on complex, real part

Average Error: 31.3 → 18.6

Time: 1.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.83181885032133734 \cdot 10^{55}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.88566167285968699 \cdot 10^{-297}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.32569577851814611 \cdot 10^{94}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double f(double re, double im) {
        double r73554 = re;
        double r73555 = r73554 * r73554;
        double r73556 = im;
        double r73557 = r73556 * r73556;
        double r73558 = r73555 + r73557;
        double r73559 = sqrt(r73558);
        double r73560 = log(r73559);
        return r73560;
}

↓

double f(double re, double im) {
        double r73561 = re;
        double r73562 = -5.831818850321337e+55;
        bool r73563 = r73561 <= r73562;
        double r73564 = -1.0;
        double r73565 = r73564 * r73561;
        double r73566 = log(r73565);
        double r73567 = -3.8099669373079583e-103;
        bool r73568 = r73561 <= r73567;
        double r73569 = r73561 * r73561;
        double r73570 = im;
        double r73571 = r73570 * r73570;
        double r73572 = r73569 + r73571;
        double r73573 = sqrt(r73572);
        double r73574 = log(r73573);
        double r73575 = 2.885661672859687e-297;
        bool r73576 = r73561 <= r73575;
        double r73577 = log(r73570);
        double r73578 = 3.325695778518146e+94;
        bool r73579 = r73561 <= r73578;
        double r73580 = log(r73561);
        double r73581 = r73579 ? r73574 : r73580;
        double r73582 = r73576 ? r73577 : r73581;
        double r73583 = r73568 ? r73574 : r73582;
        double r73584 = r73563 ? r73566 : r73583;
        return r73584;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -5.831818850321337e+55

   1. Initial program 44.4

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

   2. Taylor expanded around -inf 10.5

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

   if -5.831818850321337e+55 < re < -3.8099669373079583e-103 or 2.885661672859687e-297 < re < 3.325695778518146e+94

   1. Initial program 19.4

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

   if -3.8099669373079583e-103 < re < 2.885661672859687e-297

   1. Initial program 28.5

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

   2. Taylor expanded around 0 35.8

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

   if 3.325695778518146e+94 < re

   1. Initial program 50.1

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

   2. Taylor expanded around inf 8.7

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

4. Final simplification 18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.83181885032133734 \cdot 10^{55}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.88566167285968699 \cdot 10^{-297}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.32569577851814611 \cdot 10^{94}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))