Average Error: 31.2 → 16.8
Time: 2.0s
Precision: binary64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1769451486882908 \cdot 10^{+113}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 4.55397686345028 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.1769451486882908 \cdot 10^{+113}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq 4.55397686345028 \cdot 10^{+93}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

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

\end{array}
(FPCore (x y) :precision binary64 (sqrt (+ (* x x) (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -1.1769451486882908e+113)
   (- x)
   (if (<= x 4.55397686345028e+93) (sqrt (+ (* x x) (* y y))) x)))
double code(double x, double y) {
	return sqrt((x * x) + (y * y));
}
↓
double code(double x, double y) {
	double tmp;
	if (x <= -1.1769451486882908e+113) {
		tmp = -x;
	} else if (x <= 4.55397686345028e+93) {
		tmp = sqrt((x * x) + (y * y));
	} else {
		tmp = x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.2
Target16.8
Herbie16.8
\[\begin{array}{l} \mathbf{if}\;x < -1.1236950826599826 \cdot 10^{+145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x < 1.116557621183362 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.17694514868829084e113

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 9.8

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

    3. Simplified9.8

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

    if -1.17694514868829084e113 < x < 4.55397686345028005e93

    1. Initial program 20.3

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

    if 4.55397686345028005e93 < x

    1. Initial program 52.2

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

    2. Taylor expanded around inf 10.1

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1769451486882908 \cdot 10^{+113}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 4.55397686345028 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.1236950826599826e+145) (- x) (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x))

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