math.abs on complex

Average Error: 31.7 → 17.4

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.688721359903120564949235071844216306814 \cdot 10^{100}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.314940489788706082274641758508570148355 \cdot 10^{92}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r1501584 = re;
        double r1501585 = r1501584 * r1501584;
        double r1501586 = im;
        double r1501587 = r1501586 * r1501586;
        double r1501588 = r1501585 + r1501587;
        double r1501589 = sqrt(r1501588);
        return r1501589;
}

↓

double f(double re, double im) {
        double r1501590 = re;
        double r1501591 = -1.6887213599031206e+100;
        bool r1501592 = r1501590 <= r1501591;
        double r1501593 = -r1501590;
        double r1501594 = 4.314940489788706e+92;
        bool r1501595 = r1501590 <= r1501594;
        double r1501596 = im;
        double r1501597 = r1501596 * r1501596;
        double r1501598 = r1501590 * r1501590;
        double r1501599 = r1501597 + r1501598;
        double r1501600 = sqrt(r1501599);
        double r1501601 = r1501595 ? r1501600 : r1501590;
        double r1501602 = r1501592 ? r1501593 : r1501601;
        return r1501602;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -1.6887213599031206e+100

   1. Initial program 51.8

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

   2. Taylor expanded around -inf 10.5

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

   3. Simplified 10.5

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

   if -1.6887213599031206e+100 < re < 4.314940489788706e+92

   1. Initial program 21.3

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

   if 4.314940489788706e+92 < re

   1. Initial program 50.0

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

   2. Taylor expanded around inf 10.2

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

4. Final simplification 17.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.688721359903120564949235071844216306814 \cdot 10^{100}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.314940489788706082274641758508570148355 \cdot 10^{92}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))