Average Error: 31.1 → 6.6
Time: 2.9s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -6.031736331661589 \cdot 10^{+101}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -3.2741935890205473 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y + 0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -6.031736331661589 \cdot 10^{+101}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq -3.2741935890205473 \cdot 10^{-158}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{else}:\\
\;\;\;\;y + 0.5 \cdot \frac{x}{\frac{y}{x}}\\

\end{array}
(FPCore (x y) :precision binary64 (sqrt (+ (* x x) (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -6.031736331661589e+101)
   (- x)
   (if (<= x -3.2741935890205473e-158)
     (sqrt (+ (* x x) (* y y)))
     (+ y (* 0.5 (/ x (/ y x)))))))
double code(double x, double y) {
	return sqrt((x * x) + (y * y));
}
↓
double code(double x, double y) {
	double tmp;
	if (x <= -6.031736331661589e+101) {
		tmp = -x;
	} else if (x <= -3.2741935890205473e-158) {
		tmp = sqrt((x * x) + (y * y));
	} else {
		tmp = y + (0.5 * (x / (y / 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

Original31.1
Target16.8
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;x < -1.1236950826599826 \cdot 10^{+145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x < 1.116557621183362 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -6.031736331661589e101

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 5.0

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

    3. Simplified5.0

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

    if -6.031736331661589e101 < x < -3.2741935890205473e-158

    1. Initial program 11.1

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

    if -3.2741935890205473e-158 < x

    1. Initial program 31.2

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

    2. Taylor expanded around 0 7.1

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y} + y}\]

    3. Simplified7.1

      \[\leadsto \color{blue}{y + 0.5 \cdot \frac{x \cdot x}{y}}\]
    4. Using strategy rm

    5. Applied associate-/l*_binary64_232114.4

      \[\leadsto y + 0.5 \cdot \color{blue}{\frac{x}{\frac{y}{x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.031736331661589 \cdot 10^{+101}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -3.2741935890205473 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y + 0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))