math.abs on complex

Average Error: 29.6 → 16.8

Time: 8.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.3773968973578107 \cdot 10^{+162}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.5108781627056909 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r2149049 = re;
        double r2149050 = r2149049 * r2149049;
        double r2149051 = im;
        double r2149052 = r2149051 * r2149051;
        double r2149053 = r2149050 + r2149052;
        double r2149054 = sqrt(r2149053);
        return r2149054;
}

↓

double f(double re, double im) {
        double r2149055 = re;
        double r2149056 = -1.3773968973578107e+162;
        bool r2149057 = r2149055 <= r2149056;
        double r2149058 = -r2149055;
        double r2149059 = 1.5108781627056909e+137;
        bool r2149060 = r2149055 <= r2149059;
        double r2149061 = im;
        double r2149062 = r2149061 * r2149061;
        double r2149063 = r2149055 * r2149055;
        double r2149064 = r2149062 + r2149063;
        double r2149065 = sqrt(r2149064);
        double r2149066 = r2149060 ? r2149065 : r2149055;
        double r2149067 = r2149057 ? r2149058 : r2149066;
        return r2149067;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -1.3773968973578107e+162

   1. Initial program 59.2

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

   2. Taylor expanded around -inf 7.9

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

   3. Simplified 7.9

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

   if -1.3773968973578107e+162 < re < 1.5108781627056909e+137

   1. Initial program 19.7

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

   if 1.5108781627056909e+137 < re

   1. Initial program 54.7

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

   2. Taylor expanded around inf 9.6

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

4. Final simplification 16.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3773968973578107 \cdot 10^{+162}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.5108781627056909 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))