Average Error: 31.9 → 7.8
Time: 1.8s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq 1.1776961076709698 \cdot 10^{-158}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 9.723511542146676 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq 5.577535844625118 \cdot 10^{+41}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq 1.1776961076709698 \cdot 10^{-158}:\\
\;\;\;\;-x\\

\mathbf{elif}\;y \leq 9.723511542146676 \cdot 10^{+37}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \leq 5.577535844625118 \cdot 10^{+41}:\\
\;\;\;\;-x\\

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

\end{array}
(FPCore (x y) :precision binary64 (sqrt (+ (* x x) (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= y 1.1776961076709698e-158)
   (- x)
   (if (<= y 9.723511542146676e+37)
     (sqrt (+ (* x x) (* y y)))
     (if (<= y 5.577535844625118e+41) (- x) y))))
double code(double x, double y) {
	return sqrt((x * x) + (y * y));
}
↓
double code(double x, double y) {
	double tmp;
	if (y <= 1.1776961076709698e-158) {
		tmp = -x;
	} else if (y <= 9.723511542146676e+37) {
		tmp = sqrt((x * x) + (y * y));
	} else if (y <= 5.577535844625118e+41) {
		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

Original31.9
Target17.3
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 y < 1.1776961076709698e-158 or 9.7235115421466756e37 < y < 5.57753584462511827e41

    1. Initial program 32.8

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

    2. Taylor expanded around -inf 5.7

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

    if 1.1776961076709698e-158 < y < 9.7235115421466756e37

    1. Initial program 11.3

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

    if 5.57753584462511827e41 < y

    1. Initial program 42.8

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

    2. Taylor expanded around 0 7.9

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1776961076709698 \cdot 10^{-158}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 9.723511542146676 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq 5.577535844625118 \cdot 10^{+41}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array}\]

Reproduce

herbie shell --seed 2021093 
(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))))