Average Error: 31.4 → 19.0
Time: 4.9s
Precision: 64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.05978324146926776621503694441833231193 \cdot 10^{-253}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 3.012224090936350650107808168583637972217 \cdot 10^{56}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;x \le 1.05978324146926776621503694441833231193 \cdot 10^{-253}:\\
\;\;\;\;y\\

\mathbf{elif}\;x \le 3.012224090936350650107808168583637972217 \cdot 10^{56}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

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

\end{array}
double f(double x, double y) {
        double r502656 = x;
        double r502657 = r502656 * r502656;
        double r502658 = y;
        double r502659 = r502658 * r502658;
        double r502660 = r502657 + r502659;
        double r502661 = sqrt(r502660);
        return r502661;
}
↓
double f(double x, double y) {
        double r502662 = x;
        double r502663 = -5.330091552844717e+114;
        bool r502664 = r502662 <= r502663;
        double r502665 = -r502662;
        double r502666 = -4.2156616274993736e-144;
        bool r502667 = r502662 <= r502666;
        double r502668 = r502662 * r502662;
        double r502669 = y;
        double r502670 = r502669 * r502669;
        double r502671 = r502668 + r502670;
        double r502672 = sqrt(r502671);
        double r502673 = 1.0597832414692678e-253;
        bool r502674 = r502662 <= r502673;
        double r502675 = 3.0122240909363507e+56;
        bool r502676 = r502662 <= r502675;
        double r502677 = r502676 ? r502672 : r502662;
        double r502678 = r502674 ? r502669 : r502677;
        double r502679 = r502667 ? r502672 : r502678;
        double r502680 = r502664 ? r502665 : r502679;
        return r502680;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.4
Target17.7
Herbie19.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659982632437974301616192301785 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.116557621183362039388201959321597704512 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -5.330091552844717e+114

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 8.7

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

    3. Simplified8.7

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

    if -5.330091552844717e+114 < x < -4.2156616274993736e-144 or 1.0597832414692678e-253 < x < 3.0122240909363507e+56

    1. Initial program 18.7

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

    if -4.2156616274993736e-144 < x < 1.0597832414692678e-253

    1. Initial program 30.2

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

    2. Taylor expanded around 0 35.6

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

    if 3.0122240909363507e+56 < x

    1. Initial program 44.3

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

    2. Taylor expanded around inf 12.9

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

  4. Final simplification19.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.05978324146926776621503694441833231193 \cdot 10^{-253}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 3.012224090936350650107808168583637972217 \cdot 10^{56}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))