Average Error: 31.2 → 7.8
Time: 1.6s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.8823818060423687 \cdot 10^{+93}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -8.295868111085673 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \leq -1.7085030248267635 \cdot 10^{-189}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.8823818060423687 \cdot 10^{+93}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq -8.295868111085673 \cdot 10^{-165}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;x \leq -1.7085030248267635 \cdot 10^{-189}:\\
\;\;\;\;-x\\

\mathbf{else}:\\
\;\;\;\;y\\

\end{array}
(FPCore (x y) :precision binary64 (sqrt (+ (* x x) (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -1.8823818060423687e+93)
   (- x)
   (if (<= x -8.295868111085673e-165)
     (sqrt (+ (* x x) (* y y)))
     (if (<= x -1.7085030248267635e-189) (- x) y))))
double code(double x, double y) {
	return sqrt((x * x) + (y * y));
}
↓
double code(double x, double y) {
	double tmp;
	if (x <= -1.8823818060423687e+93) {
		tmp = -x;
	} else if (x <= -8.295868111085673e-165) {
		tmp = sqrt((x * x) + (y * y));
	} else if (x <= -1.7085030248267635e-189) {
		tmp = -x;
	} else {
		tmp = y;
	}
	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
Target17.5
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;x < -1.1236950826599826 \cdot 10^{+145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x < 1.116557621183362 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.88238180604236865e93 or -8.29586811108567327e-165 < x < -1.70850302482676351e-189

    1. Initial program 48.3

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

    2. Taylor expanded around -inf 8.6

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

    3. Simplified8.6

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

    if -1.88238180604236865e93 < x < -8.29586811108567327e-165

    1. Initial program 11.7

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

    if -1.70850302482676351e-189 < x

    1. Initial program 31.5

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

    2. Taylor expanded around 0 3.9

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8823818060423687 \cdot 10^{+93}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -8.295868111085673 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \leq -1.7085030248267635 \cdot 10^{-189}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(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))))