Average Error: 31.8 → 18.0
Time: 1.4s
Precision: binary64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4.864167289730826 \cdot 10^{+136}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -2.4704911317770355 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \leq -1.6020683207984553 \cdot 10^{-95}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 1.1635286815108139 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4.864167289730826 \cdot 10^{+136}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq -2.4704911317770355 \cdot 10^{-70}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;x \leq -1.6020683207984553 \cdot 10^{-95}:\\
\;\;\;\;y\\

\mathbf{elif}\;x \leq 1.1635286815108139 \cdot 10^{+92}:\\
\;\;\;\;\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 -4.864167289730826e+136)
   (- x)
   (if (<= x -2.4704911317770355e-70)
     (sqrt (+ (* x x) (* y y)))
     (if (<= x -1.6020683207984553e-95)
       y
       (if (<= x 1.1635286815108139e+92) (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 <= -4.864167289730826e+136)) {
		tmp = ((double) -(x));
	} else {
		double tmp_1;
		if ((x <= -2.4704911317770355e-70)) {
			tmp_1 = ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
		} else {
			double tmp_2;
			if ((x <= -1.6020683207984553e-95)) {
				tmp_2 = y;
			} else {
				double tmp_3;
				if ((x <= 1.1635286815108139e+92)) {
					tmp_3 = ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
				} else {
					tmp_3 = x;
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		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

Original31.8
Target17.6
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 4 regimes

  2. if x < -4.8641672897308265e136

    1. Initial program Error: 59.4 bits

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

    2. Taylor expanded around -inf Error: 9.6 bits

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

    3. SimplifiedError: 9.6 bits

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

    if -4.8641672897308265e136 < x < -2.4704911317770355e-70 or -1.6020683207984553e-95 < x < 1.1635286815108139e92

    1. Initial program Error: 21.0 bits

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

    if -2.4704911317770355e-70 < x < -1.6020683207984553e-95

    1. Initial program Error: 20.5 bits

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

    2. Taylor expanded around 0 Error: 39.2 bits

      \[\leadsto \color{blue}{y}\]

    if 1.1635286815108139e92 < x

    1. Initial program Error: 49.8 bits

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

    2. Taylor expanded around inf Error: 11.6 bits

      \[\leadsto \color{blue}{x}\]
  3. Recombined 4 regimes into one program.

  4. Final simplificationError: 18.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.864167289730826 \cdot 10^{+136}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -2.4704911317770355 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \leq -1.6020683207984553 \cdot 10^{-95}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 1.1635286815108139 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))