math.abs on complex

Average Error: 29.4 → 16.6

Time: 2.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.1292868428778451 \cdot 10^{+139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.0853955874561044 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.2456535811590808 \cdot 10^{-300}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.8099256402278783 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 5.721648594770613 \cdot 10^{-217}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.4073759586339516 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r862814 = re;
        double r862815 = r862814 * r862814;
        double r862816 = im;
        double r862817 = r862816 * r862816;
        double r862818 = r862815 + r862817;
        double r862819 = sqrt(r862818);
        return r862819;
}

↓

double f(double re, double im) {
        double r862820 = re;
        double r862821 = -1.1292868428778451e+139;
        bool r862822 = r862820 <= r862821;
        double r862823 = -r862820;
        double r862824 = -1.0853955874561044e-276;
        bool r862825 = r862820 <= r862824;
        double r862826 = im;
        double r862827 = r862826 * r862826;
        double r862828 = r862820 * r862820;
        double r862829 = r862827 + r862828;
        double r862830 = sqrt(r862829);
        double r862831 = 1.2456535811590808e-300;
        bool r862832 = r862820 <= r862831;
        double r862833 = 1.8099256402278783e-246;
        bool r862834 = r862820 <= r862833;
        double r862835 = 5.721648594770613e-217;
        bool r862836 = r862820 <= r862835;
        double r862837 = 2.4073759586339516e+149;
        bool r862838 = r862820 <= r862837;
        double r862839 = r862838 ? r862830 : r862820;
        double r862840 = r862836 ? r862826 : r862839;
        double r862841 = r862834 ? r862830 : r862840;
        double r862842 = r862832 ? r862826 : r862841;
        double r862843 = r862825 ? r862830 : r862842;
        double r862844 = r862822 ? r862823 : r862843;
        return r862844;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.1292868428778451e+139

   1. Initial program 55.4

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

   2. Taylor expanded around -inf 8.5

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

   3. Simplified 8.5

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

   if -1.1292868428778451e+139 < re < -1.0853955874561044e-276 or 1.2456535811590808e-300 < re < 1.8099256402278783e-246 or 5.721648594770613e-217 < re < 2.4073759586339516e+149

   1. Initial program 18.8

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

   if -1.0853955874561044e-276 < re < 1.2456535811590808e-300 or 1.8099256402278783e-246 < re < 5.721648594770613e-217

   1. Initial program 32.1

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

   2. Taylor expanded around 0 32.9

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

   if 2.4073759586339516e+149 < re

   1. Initial program 57.9

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

   2. Taylor expanded around inf 6.8

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

4. Final simplification 16.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1292868428778451 \cdot 10^{+139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.0853955874561044 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.2456535811590808 \cdot 10^{-300}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.8099256402278783 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 5.721648594770613 \cdot 10^{-217}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.4073759586339516 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))