math.abs on complex

Average Error: 31.6 → 17.1

Time: 20.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -8.039547558546271631628934502111036051442 \cdot 10^{104}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -3.240034389533803828723034610715031267776 \cdot 10^{-262}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 2.715396324398381567593668375558902625339 \cdot 10^{-222}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.683247713446899747987853789973400736076 \cdot 10^{123}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r1626127 = re;
        double r1626128 = r1626127 * r1626127;
        double r1626129 = im;
        double r1626130 = r1626129 * r1626129;
        double r1626131 = r1626128 + r1626130;
        double r1626132 = sqrt(r1626131);
        return r1626132;
}

↓

double f(double re, double im) {
        double r1626133 = re;
        double r1626134 = -8.039547558546272e+104;
        bool r1626135 = r1626133 <= r1626134;
        double r1626136 = -r1626133;
        double r1626137 = -3.240034389533804e-262;
        bool r1626138 = r1626133 <= r1626137;
        double r1626139 = im;
        double r1626140 = r1626139 * r1626139;
        double r1626141 = r1626133 * r1626133;
        double r1626142 = r1626140 + r1626141;
        double r1626143 = sqrt(r1626142);
        double r1626144 = 2.7153963243983816e-222;
        bool r1626145 = r1626133 <= r1626144;
        double r1626146 = 2.6832477134469e+123;
        bool r1626147 = r1626133 <= r1626146;
        double r1626148 = r1626147 ? r1626143 : r1626133;
        double r1626149 = r1626145 ? r1626139 : r1626148;
        double r1626150 = r1626138 ? r1626143 : r1626149;
        double r1626151 = r1626135 ? r1626136 : r1626150;
        return r1626151;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -8.039547558546272e+104

   1. Initial program 52.4

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

   2. Taylor expanded around -inf 9.5

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

   3. Simplified 9.5

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

   if -8.039547558546272e+104 < re < -3.240034389533804e-262 or 2.7153963243983816e-222 < re < 2.6832477134469e+123

   1. Initial program 19.0

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

   if -3.240034389533804e-262 < re < 2.7153963243983816e-222

   1. Initial program 32.1

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

   2. Taylor expanded around 0 31.4

      \[\leadsto \color{blue}{im}\]

   if 2.6832477134469e+123 < re

   1. Initial program 55.3

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

   2. Taylor expanded around inf 8.3

      \[\leadsto \color{blue}{re}\]
3. Recombined 4 regimes into one program.

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.039547558546271631628934502111036051442 \cdot 10^{104}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -3.240034389533803828723034610715031267776 \cdot 10^{-262}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 2.715396324398381567593668375558902625339 \cdot 10^{-222}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.683247713446899747987853789973400736076 \cdot 10^{123}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))