Average Error: 21.2 → 0.1
Time: 1.8s
Precision: binary64
\[\sqrt{x \cdot x + y} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -6.309494455453091 \cdot 10^{+153}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.2583755260511737 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -6.309494455453091 \cdot 10^{+153}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq 2.2583755260511737 \cdot 10^{+115}:\\
\;\;\;\;\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 -6.309494455453091e+153)
   (- x)
   (if (<= x 2.2583755260511737e+115)
     (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 <= -6.309494455453091e+153) {
		tmp = -x;
	} else if (x <= 2.2583755260511737e+115) {
		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

Original21.2
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x < -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x < 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -6.3094944554530911e153

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 0

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

    3. Simplified0

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

    if -6.3094944554530911e153 < x < 2.25837552605117373e115

    1. Initial program 0.0

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

    if 2.25837552605117373e115 < x

    1. Initial program 51.7

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.309494455453091 \cdot 10^{+153}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.2583755260511737 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(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)))