Average Error: 32.0 → 18.4
Time: 1.9s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.9648958467837301 \cdot 10^{102}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 3.49168277985170575 \cdot 10^{70}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.9648958467837301 \cdot 10^{102}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 3.49168277985170575 \cdot 10^{70}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;re\\

\end{array}
double f(double re, double im) {
        double r103826 = re;
        double r103827 = r103826 * r103826;
        double r103828 = im;
        double r103829 = r103828 * r103828;
        double r103830 = r103827 + r103829;
        double r103831 = sqrt(r103830);
        return r103831;
}
↓
double f(double re, double im) {
        double r103832 = re;
        double r103833 = -5.96489584678373e+102;
        bool r103834 = r103832 <= r103833;
        double r103835 = -1.0;
        double r103836 = r103835 * r103832;
        double r103837 = 3.491682779851706e+70;
        bool r103838 = r103832 <= r103837;
        double r103839 = r103832 * r103832;
        double r103840 = im;
        double r103841 = r103840 * r103840;
        double r103842 = r103839 + r103841;
        double r103843 = sqrt(r103842);
        double r103844 = r103838 ? r103843 : r103832;
        double r103845 = r103834 ? r103836 : r103844;
        return r103845;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -5.96489584678373e+102

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 10.2

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

    if -5.96489584678373e+102 < re < 3.491682779851706e+70

    1. Initial program 22.2

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

    if 3.491682779851706e+70 < re

    1. Initial program 47.4

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

    2. Taylor expanded around inf 12.5

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

  4. Final simplification18.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.9648958467837301 \cdot 10^{102}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 3.49168277985170575 \cdot 10^{70}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))