Average Error: 20.5 → 0.2
Time: 2.7s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.36455936512260061 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 7.94205206826031504 \cdot 10^{104}:\\ \;\;\;\;\sqrt{1 \cdot \mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.36455936512260061 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 7.94205206826031504 \cdot 10^{104}:\\
\;\;\;\;\sqrt{1 \cdot \mathsf{fma}\left(x, x, y\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\

\end{array}
double code(double x, double y) {
	return ((double) sqrt(((double) (((double) (x * x)) + y))));
}
↓
double code(double x, double y) {
	double VAR;
	if ((x <= -1.3645593651226006e+154)) {
		VAR = ((double) -(((double) (x + ((double) (0.5 * ((double) (y / x))))))));
	} else {
		double VAR_1;
		if ((x <= 7.942052068260315e+104)) {
			VAR_1 = ((double) sqrt(((double) (1.0 * ((double) fma(x, x, y))))));
		} else {
			VAR_1 = ((double) fma(0.5, ((double) (y / x)), 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

Original20.5
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3645593651226006e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

      \[\leadsto \color{blue}{-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)}\]

    if -1.3645593651226006e+154 < x < 7.942052068260315e+104

    1. Initial program 0.0

      \[\sqrt{x \cdot x + y}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.0

      \[\leadsto \sqrt{x \cdot x + \color{blue}{1 \cdot y}}\]

    4. Applied *-un-lft-identity0.0

      \[\leadsto \sqrt{\color{blue}{1 \cdot \left(x \cdot x\right)} + 1 \cdot y}\]

    5. Applied distribute-lft-out0.0

      \[\leadsto \sqrt{\color{blue}{1 \cdot \left(x \cdot x + y\right)}}\]

    6. Simplified0.0

      \[\leadsto \sqrt{1 \cdot \color{blue}{\mathsf{fma}\left(x, x, y\right)}}\]

    if 7.942052068260315e+104 < x

    1. Initial program 48.4

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

    2. Taylor expanded around inf 0.7

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \frac{y}{x}}\]

    3. Simplified0.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.36455936512260061 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 7.94205206826031504 \cdot 10^{104}:\\ \;\;\;\;\sqrt{1 \cdot \mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(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)))