Average Error: 21.5 → 0.7
Time: 1.5s
Precision: binary64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.3035121163067228 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.7677596047938106 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.3035121163067228 \cdot 10^{+154}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq 1.7677596047938106 \cdot 10^{+27}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
(FPCore (x y) :precision binary64 (sqrt (+ (* x x) y)))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -1.3035121163067228e+154)
   (- x)
   (if (<= x 1.7677596047938106e+27)
     (sqrt (+ (* x x) y))
     (+ x (* 0.5 (/ y x))))))
double code(double x, double y) {
	return sqrt((x * x) + y);
}
↓
double code(double x, double y) {
	double tmp;
	if (x <= -1.3035121163067228e+154) {
		tmp = -x;
	} else if (x <= 1.7677596047938106e+27) {
		tmp = sqrt((x * x) + y);
	} else {
		tmp = x + (0.5 * (y / x));
	}
	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

Original21.5
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x < -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x < 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.30351211630672276e154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0.0

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

    3. Simplified0.0

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

    if -1.30351211630672276e154 < x < 1.76775960479381059e27

    1. Initial program 0.0

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

    if 1.76775960479381059e27 < x

    1. Initial program 36.9

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

    2. Taylor expanded around inf 2.5

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

    3. Simplified2.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3035121163067228 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.7677596047938106 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(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)))