Average Error: 32.1 → 17.8
Time: 4.5s
Precision: 64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.219332295965777137041720193068407814529 \cdot 10^{82}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -3.743447547042940916879606925039648794356 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 4.769025653725654548986941102749859144285 \cdot 10^{-305}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 2.419375064734749687649336536979338940651 \cdot 10^{132}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.219332295965777137041720193068407814529 \cdot 10^{82}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \le -3.743447547042940916879606925039648794356 \cdot 10^{-217}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;x \le 4.769025653725654548986941102749859144285 \cdot 10^{-305}:\\
\;\;\;\;y\\

\mathbf{elif}\;x \le 2.419375064734749687649336536979338940651 \cdot 10^{132}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
double f(double x, double y) {
        double r486652 = x;
        double r486653 = r486652 * r486652;
        double r486654 = y;
        double r486655 = r486654 * r486654;
        double r486656 = r486653 + r486655;
        double r486657 = sqrt(r486656);
        return r486657;
}
↓
double f(double x, double y) {
        double r486658 = x;
        double r486659 = -4.219332295965777e+82;
        bool r486660 = r486658 <= r486659;
        double r486661 = -r486658;
        double r486662 = -3.743447547042941e-217;
        bool r486663 = r486658 <= r486662;
        double r486664 = r486658 * r486658;
        double r486665 = y;
        double r486666 = r486665 * r486665;
        double r486667 = r486664 + r486666;
        double r486668 = sqrt(r486667);
        double r486669 = 4.769025653725655e-305;
        bool r486670 = r486658 <= r486669;
        double r486671 = 2.4193750647347497e+132;
        bool r486672 = r486658 <= r486671;
        double r486673 = r486672 ? r486668 : r486658;
        double r486674 = r486670 ? r486665 : r486673;
        double r486675 = r486663 ? r486668 : r486674;
        double r486676 = r486660 ? r486661 : r486675;
        return r486676;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.1
Target17.8
Herbie17.8
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659982632437974301616192301785 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.116557621183362039388201959321597704512 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -4.219332295965777e+82

    1. Initial program 49.1

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around -inf 11.7

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

    3. Simplified11.7

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

    if -4.219332295965777e+82 < x < -3.743447547042941e-217 or 4.769025653725655e-305 < x < 2.4193750647347497e+132

    1. Initial program 20.0

      \[\sqrt{x \cdot x + y \cdot y}\]

    if -3.743447547042941e-217 < x < 4.769025653725655e-305

    1. Initial program 33.9

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around 0 33.5

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

    if 2.4193750647347497e+132 < x

    1. Initial program 58.5

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around inf 8.4

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.219332295965777137041720193068407814529 \cdot 10^{82}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -3.743447547042940916879606925039648794356 \cdot 10^{-217}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 4.769025653725654548986941102749859144285 \cdot 10^{-305}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 2.419375064734749687649336536979338940651 \cdot 10^{132}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.123695082659983e+145) (- x) (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x))

  (sqrt (+ (* x x) (* y y))))