math.abs on complex

Average Error: 29.3 → 16.1

Time: 3.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.835831488246852 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.485304045568738 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r1479777 = re;
        double r1479778 = r1479777 * r1479777;
        double r1479779 = im;
        double r1479780 = r1479779 * r1479779;
        double r1479781 = r1479778 + r1479780;
        double r1479782 = sqrt(r1479781);
        return r1479782;
}

↓

double f(double re, double im) {
        double r1479783 = re;
        double r1479784 = -2.835831488246852e+153;
        bool r1479785 = r1479783 <= r1479784;
        double r1479786 = -r1479783;
        double r1479787 = 5.485304045568738e+122;
        bool r1479788 = r1479783 <= r1479787;
        double r1479789 = im;
        double r1479790 = r1479789 * r1479789;
        double r1479791 = r1479783 * r1479783;
        double r1479792 = r1479790 + r1479791;
        double r1479793 = sqrt(r1479792);
        double r1479794 = r1479788 ? r1479793 : r1479783;
        double r1479795 = r1479785 ? r1479786 : r1479794;
        return r1479795;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -2.835831488246852e+153

   1. Initial program 59.3

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

   2. Taylor expanded around -inf 7.1

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

   3. Simplified 7.1

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

   if -2.835831488246852e+153 < re < 5.485304045568738e+122

   1. Initial program 19.2

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

   if 5.485304045568738e+122 < re

   1. Initial program 53.0

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

   2. Taylor expanded around inf 8.6

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

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.835831488246852 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.485304045568738 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))