math.abs on complex

Average Error: 32.0 → 18.1

Time: 3.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.488419020361885603121058718518165058597 \cdot 10^{90}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 321983510253822902678712615891538695684100:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r59165 = re;
        double r59166 = r59165 * r59165;
        double r59167 = im;
        double r59168 = r59167 * r59167;
        double r59169 = r59166 + r59168;
        double r59170 = sqrt(r59169);
        return r59170;
}

↓

double f(double re, double im) {
        double r59171 = re;
        double r59172 = -3.4884190203618856e+90;
        bool r59173 = r59171 <= r59172;
        double r59174 = -1.0;
        double r59175 = r59174 * r59171;
        double r59176 = 3.219835102538229e+41;
        bool r59177 = r59171 <= r59176;
        double r59178 = r59171 * r59171;
        double r59179 = im;
        double r59180 = r59179 * r59179;
        double r59181 = r59178 + r59180;
        double r59182 = sqrt(r59181);
        double r59183 = r59177 ? r59182 : r59171;
        double r59184 = r59173 ? r59175 : r59183;
        return r59184;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -3.4884190203618856e+90

   1. Initial program 50.6

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

   2. Taylor expanded around -inf 10.4

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

   if -3.4884190203618856e+90 < re < 3.219835102538229e+41

   1. Initial program 21.9

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

   if 3.219835102538229e+41 < re

   1. Initial program 45.5

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

   2. Taylor expanded around inf 13.7

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

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.488419020361885603121058718518165058597 \cdot 10^{90}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 321983510253822902678712615891538695684100:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))