math.abs on complex

Average Error: 29.9 → 17.1

Time: 7.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -7.68112807196736 \cdot 10^{+158}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.574733061767586 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r1545176 = re;
        double r1545177 = r1545176 * r1545176;
        double r1545178 = im;
        double r1545179 = r1545178 * r1545178;
        double r1545180 = r1545177 + r1545179;
        double r1545181 = sqrt(r1545180);
        return r1545181;
}

↓

double f(double re, double im) {
        double r1545182 = re;
        double r1545183 = -7.68112807196736e+158;
        bool r1545184 = r1545182 <= r1545183;
        double r1545185 = -r1545182;
        double r1545186 = 1.574733061767586e+151;
        bool r1545187 = r1545182 <= r1545186;
        double r1545188 = im;
        double r1545189 = r1545188 * r1545188;
        double r1545190 = r1545182 * r1545182;
        double r1545191 = r1545189 + r1545190;
        double r1545192 = sqrt(r1545191);
        double r1545193 = r1545187 ? r1545192 : r1545182;
        double r1545194 = r1545184 ? r1545185 : r1545193;
        return r1545194;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -7.68112807196736e+158

   1. Initial program 59.3

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

   2. Taylor expanded around -inf 6.7

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

   3. Simplified 6.7

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

   if -7.68112807196736e+158 < re < 1.574733061767586e+151

   1. Initial program 20.2

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

   if 1.574733061767586e+151 < re

   1. Initial program 58.7

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

   2. Taylor expanded around inf 8.4

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.68112807196736 \cdot 10^{+158}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.574733061767586 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))