math.abs on complex

Average Error: 29.3 → 16.1

Time: 2.6s

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 r1233296 = re;
        double r1233297 = r1233296 * r1233296;
        double r1233298 = im;
        double r1233299 = r1233298 * r1233298;
        double r1233300 = r1233297 + r1233299;
        double r1233301 = sqrt(r1233300);
        return r1233301;
}

↓

double f(double re, double im) {
        double r1233302 = re;
        double r1233303 = -2.835831488246852e+153;
        bool r1233304 = r1233302 <= r1233303;
        double r1233305 = -r1233302;
        double r1233306 = 5.485304045568738e+122;
        bool r1233307 = r1233302 <= r1233306;
        double r1233308 = im;
        double r1233309 = r1233308 * r1233308;
        double r1233310 = r1233302 * r1233302;
        double r1233311 = r1233309 + r1233310;
        double r1233312 = sqrt(r1233311);
        double r1233313 = r1233307 ? r1233312 : r1233302;
        double r1233314 = r1233304 ? r1233305 : r1233313;
        return r1233314;
}

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))))