Average Error: 31.5 → 18.0
Time: 1.6s
Precision: binary64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2.8473355353767562 \cdot 10^{+116}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.3232582542635886 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2.8473355353767562 \cdot 10^{+116}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq 2.3232582542635886 \cdot 10^{+20}:\\
\;\;\;\;\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 -2.8473355353767562e+116)
   (- x)
   (if (<= x 2.3232582542635886e+20) (sqrt (+ (* x x) (* y y))) x)))
double code(double x, double y) {
	return ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
}
↓
double code(double x, double y) {
	double tmp;
	if ((x <= -2.8473355353767562e+116)) {
		tmp = ((double) -(x));
	} else {
		double tmp_1;
		if ((x <= 2.3232582542635886e+20)) {
			tmp_1 = ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
		} else {
			tmp_1 = x;
		}
		tmp = tmp_1;
	}
	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

Original31.5
Target17.4
Herbie18.0
\[\begin{array}{l} \mathbf{if}\;x < -1.1236950826599826 \cdot 10^{+145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x < 1.116557621183362 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.8473355353767562e116

    1. Initial program Error: 54.7 bits

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

    2. Taylor expanded around -inf Error: 8.6 bits

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

    3. SimplifiedError: 8.6 bits

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

    if -2.8473355353767562e116 < x < 232325825426358860000

    1. Initial program Error: 22.0 bits

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

    if 232325825426358860000 < x

    1. Initial program Error: 40.4 bits

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

    2. Taylor expanded around inf Error: 13.8 bits

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

  4. Final simplificationError: 18.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8473355353767562 \cdot 10^{+116}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.3232582542635886 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))))