Average Error: 21.6 → 0.3
Time: 1.4s
Precision: binary64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.369012744344831 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{elif}\;x \leq 3.3449376977611315 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.369012744344831 \cdot 10^{+154}:\\
\;\;\;\;y \cdot \frac{-0.5}{x} - x\\

\mathbf{elif}\;x \leq 3.3449376977611315 \cdot 10^{+83}:\\
\;\;\;\;\sqrt{y + x \cdot x}\\

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

\end{array}
(FPCore (x y) :precision binary64 (sqrt (+ (* x x) y)))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -1.369012744344831e+154)
   (- (* y (/ -0.5 x)) x)
   (if (<= x 3.3449376977611315e+83)
     (sqrt (+ y (* x x)))
     (+ x (* y (/ 0.5 x))))))
double code(double x, double y) {
	return ((double) sqrt(((double) (((double) (x * x)) + y))));
}
↓
double code(double x, double y) {
	double VAR;
	if ((x <= -1.369012744344831e+154)) {
		VAR = ((double) (((double) (y * (-0.5 / x))) - x));
	} else {
		double VAR_1;
		if ((x <= 3.3449376977611315e+83)) {
			VAR_1 = ((double) sqrt(((double) (y + ((double) (x * x))))));
		} else {
			VAR_1 = ((double) (x + ((double) (y * (0.5 / x)))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.6
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x < -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x < 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.369012744344831e154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

      \[\leadsto \color{blue}{-\left(x + 0.5 \cdot \frac{y}{x}\right)}\]

    3. Simplified0

      \[\leadsto \color{blue}{y \cdot \frac{-0.5}{x} - x}\]

    if -1.369012744344831e154 < x < 3.3449376977611315e83

    1. Initial program 0.0

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

    if 3.3449376977611315e83 < x

    1. Initial program 45.1

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

    2. Taylor expanded around inf 1.1

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

    3. Simplified1.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.369012744344831 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{elif}\;x \leq 3.3449376977611315 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e+153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.582399551122541e+57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

  (sqrt (+ (* x x) y)))