math.abs on complex

Average Error: 29.5 → 16.4

Time: 17.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.114939147891607 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 7.2373920079010435 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r1680262 = re;
        double r1680263 = r1680262 * r1680262;
        double r1680264 = im;
        double r1680265 = r1680264 * r1680264;
        double r1680266 = r1680263 + r1680265;
        double r1680267 = sqrt(r1680266);
        return r1680267;
}

↓

double f(double re, double im) {
        double r1680268 = re;
        double r1680269 = -1.114939147891607e+154;
        bool r1680270 = r1680268 <= r1680269;
        double r1680271 = -r1680268;
        double r1680272 = 7.2373920079010435e+152;
        bool r1680273 = r1680268 <= r1680272;
        double r1680274 = im;
        double r1680275 = r1680274 * r1680274;
        double r1680276 = r1680268 * r1680268;
        double r1680277 = r1680275 + r1680276;
        double r1680278 = sqrt(r1680277);
        double r1680279 = r1680273 ? r1680278 : r1680268;
        double r1680280 = r1680270 ? r1680271 : r1680279;
        return r1680280;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -1.114939147891607e+154

   1. Initial program 59.4

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

   2. Taylor expanded around -inf 6.5

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

   3. Simplified 6.5

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

   if -1.114939147891607e+154 < re < 7.2373920079010435e+152

   1. Initial program 19.5

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

   if 7.2373920079010435e+152 < re

   1. Initial program 59.1

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

   2. Taylor expanded around inf 7.4

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

4. Final simplification 16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.114939147891607 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 7.2373920079010435 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))