math.abs on complex

Average Error: 31.8 → 18.0

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.133795376408894095114882696362622107441 \cdot 10^{79}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.844559323175136428258475744554794771695 \cdot 10^{102}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r1788655 = re;
        double r1788656 = r1788655 * r1788655;
        double r1788657 = im;
        double r1788658 = r1788657 * r1788657;
        double r1788659 = r1788656 + r1788658;
        double r1788660 = sqrt(r1788659);
        return r1788660;
}

↓

double f(double re, double im) {
        double r1788661 = re;
        double r1788662 = -4.133795376408894e+79;
        bool r1788663 = r1788661 <= r1788662;
        double r1788664 = -r1788661;
        double r1788665 = 2.8445593231751364e+102;
        bool r1788666 = r1788661 <= r1788665;
        double r1788667 = im;
        double r1788668 = r1788667 * r1788667;
        double r1788669 = r1788661 * r1788661;
        double r1788670 = r1788668 + r1788669;
        double r1788671 = sqrt(r1788670);
        double r1788672 = r1788666 ? r1788671 : r1788661;
        double r1788673 = r1788663 ? r1788664 : r1788672;
        return r1788673;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -4.133795376408894e+79

   1. Initial program 49.2

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

   2. Taylor expanded around -inf 11.9

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

   3. Simplified 11.9

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

   if -4.133795376408894e+79 < re < 2.8445593231751364e+102

   1. Initial program 21.7

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

   if 2.8445593231751364e+102 < re

   1. Initial program 52.1

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

   2. Taylor expanded around inf 10.5

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.133795376408894095114882696362622107441 \cdot 10^{79}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.844559323175136428258475744554794771695 \cdot 10^{102}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))