math.abs on complex

Average Error: 29.5 → 17.2

Time: 4.3s

Precision: 64

Math

\[\sqrt{re \cdot re + im \cdot im}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.503113931025561 \cdot 10^{+161}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.604318382902631 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r2108531 = re;
        double r2108532 = r2108531 * r2108531;
        double r2108533 = im;
        double r2108534 = r2108533 * r2108533;
        double r2108535 = r2108532 + r2108534;
        double r2108536 = sqrt(r2108535);
        return r2108536;
}

↓

double f(double re, double im) {
        double r2108537 = re;
        double r2108538 = -5.503113931025561e+161;
        bool r2108539 = r2108537 <= r2108538;
        double r2108540 = -r2108537;
        double r2108541 = 5.604318382902631e+138;
        bool r2108542 = r2108537 <= r2108541;
        double r2108543 = im;
        double r2108544 = r2108543 * r2108543;
        double r2108545 = r2108537 * r2108537;
        double r2108546 = r2108544 + r2108545;
        double r2108547 = sqrt(r2108546);
        double r2108548 = r2108542 ? r2108547 : r2108537;
        double r2108549 = r2108539 ? r2108540 : r2108548;
        return r2108549;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -5.503113931025561e+161

   1. Initial program 59.3

      \[\sqrt{re \cdot re + im \cdot im}\]

   2. Taylor expanded around -inf 6.6

      \[\leadsto \color{blue}{-1 \cdot re}\]

   3. Simplified 6.6

      \[\leadsto \color{blue}{-re}\]

   if -5.503113931025561e+161 < re < 5.604318382902631e+138

   1. Initial program 20.3

      \[\sqrt{re \cdot re + im \cdot im}\]

   if 5.604318382902631e+138 < re

   1. Initial program 55.1

      \[\sqrt{re \cdot re + im \cdot im}\]

   2. Taylor expanded around inf 9.1

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

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.503113931025561 \cdot 10^{+161}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.604318382902631 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))