math.abs on complex

Average Error: 29.1 → 16.3

Time: 11.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.5255786888219323 \cdot 10^{+155}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.8950881144742005 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

C

double f(double re, double im) {
        double r1344826 = re;
        double r1344827 = r1344826 * r1344826;
        double r1344828 = im;
        double r1344829 = r1344828 * r1344828;
        double r1344830 = r1344827 + r1344829;
        double r1344831 = sqrt(r1344830);
        return r1344831;
}

↓

double f(double re, double im) {
        double r1344832 = re;
        double r1344833 = -3.5255786888219323e+155;
        bool r1344834 = r1344832 <= r1344833;
        double r1344835 = -r1344832;
        double r1344836 = 1.8950881144742005e+151;
        bool r1344837 = r1344832 <= r1344836;
        double r1344838 = im;
        double r1344839 = r1344838 * r1344838;
        double r1344840 = r1344832 * r1344832;
        double r1344841 = r1344839 + r1344840;
        double r1344842 = sqrt(r1344841);
        double r1344843 = r1344837 ? r1344842 : r1344832;
        double r1344844 = r1344834 ? r1344835 : r1344843;
        return r1344844;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -3.5255786888219323e+155

   1. Initial program 59.3

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

   2. Taylor expanded around -inf 6.3

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

   3. Simplified 6.3

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

   if -3.5255786888219323e+155 < re < 1.8950881144742005e+151

   1. Initial program 19.3

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

   if 1.8950881144742005e+151 < re

   1. Initial program 58.4

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

   2. Taylor expanded around inf 7.9

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

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.5255786888219323 \cdot 10^{+155}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.8950881144742005 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))