math.abs on complex

Average Error: 31.5 → 18.0

Time: 8.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.940119593462783780503740532731557409155 \cdot 10^{70}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.149380376403675710903737768625835202966 \cdot 10^{95}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r2083425 = re;
        double r2083426 = r2083425 * r2083425;
        double r2083427 = im;
        double r2083428 = r2083427 * r2083427;
        double r2083429 = r2083426 + r2083428;
        double r2083430 = sqrt(r2083429);
        return r2083430;
}

↓

double f(double re, double im) {
        double r2083431 = re;
        double r2083432 = -1.9401195934627838e+70;
        bool r2083433 = r2083431 <= r2083432;
        double r2083434 = -r2083431;
        double r2083435 = 1.1493803764036757e+95;
        bool r2083436 = r2083431 <= r2083435;
        double r2083437 = im;
        double r2083438 = r2083437 * r2083437;
        double r2083439 = r2083431 * r2083431;
        double r2083440 = r2083438 + r2083439;
        double r2083441 = sqrt(r2083440);
        double r2083442 = r2083436 ? r2083441 : r2083431;
        double r2083443 = r2083433 ? r2083434 : r2083442;
        return r2083443;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -1.9401195934627838e+70

   1. Initial program 46.3

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

   2. Taylor expanded around -inf 12.1

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

   3. Simplified 12.1

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

   if -1.9401195934627838e+70 < re < 1.1493803764036757e+95

   1. Initial program 21.8

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

   if 1.1493803764036757e+95 < re

   1. Initial program 49.0

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

   2. Taylor expanded around inf 11.2

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.940119593462783780503740532731557409155 \cdot 10^{70}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.149380376403675710903737768625835202966 \cdot 10^{95}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))