Average Error: 31.4 → 19.0
Time: 5.0s
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 r542974 = x;
        double r542975 = r542974 * r542974;
        double r542976 = y;
        double r542977 = r542976 * r542976;
        double r542978 = r542975 + r542977;
        double r542979 = sqrt(r542978);
        return r542979;
}
↓
double f(double x, double y) {
        double r542980 = x;
        double r542981 = -5.330091552844717e+114;
        bool r542982 = r542980 <= r542981;
        double r542983 = -r542980;
        double r542984 = -4.2156616274993736e-144;
        bool r542985 = r542980 <= r542984;
        double r542986 = r542980 * r542980;
        double r542987 = y;
        double r542988 = r542987 * r542987;
        double r542989 = r542986 + r542988;
        double r542990 = sqrt(r542989);
        double r542991 = 1.0597832414692678e-253;
        bool r542992 = r542980 <= r542991;
        double r542993 = 3.0122240909363507e+56;
        bool r542994 = r542980 <= r542993;
        double r542995 = r542994 ? r542990 : r542980;
        double r542996 = r542992 ? r542987 : r542995;
        double r542997 = r542985 ? r542990 : r542996;
        double r542998 = r542982 ? r542983 : r542997;
        return r542998;
}

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))))