Average Error: 20.2 → 5.2
Time: 2.0s
Precision: binary64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -9.82712184536510351 \cdot 10^{60}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.86052209567788248 \cdot 10^{-161} \lor \neg \left(y \le 2.06368582445499696 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x - y}{\frac{x \cdot x + y \cdot y}{y + x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -9.82712184536510351 \cdot 10^{60}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.86052209567788248 \cdot 10^{-161} \lor \neg \left(y \le 2.06368582445499696 \cdot 10^{-171}\right):\\
\;\;\;\;\frac{x - y}{\frac{x \cdot x + y \cdot y}{y + x}}\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}
double code(double x, double y) {
	return ((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))));
}
↓
double code(double x, double y) {
	double VAR;
	if ((y <= -9.827121845365104e+60)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if (((y <= -1.8605220956778825e-161) || !(y <= 2.063685824454997e-171))) {
			VAR_1 = ((double) (((double) (x - y)) / ((double) (((double) (((double) (x * x)) + ((double) (y * y)))) / ((double) (y + x))))));
		} else {
			VAR_1 = 1.0;
		}
		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

Original20.2
Target0.1
Herbie5.2
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -9.82712184536510351e60

    1. Initial program 40.1

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

    2. Taylor expanded around 0 0

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

    if -9.82712184536510351e60 < y < -1.86052209567788248e-161 or 2.06368582445499696e-171 < y

    1. Initial program 0.6

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
    2. Using strategy rm

    3. Applied associate-/l*1.2

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

    if -1.86052209567788248e-161 < y < 2.06368582445499696e-171

    1. Initial program 30.8

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

    2. Taylor expanded around inf 15.4

      \[\leadsto \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -9.82712184536510351 \cdot 10^{60}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.86052209567788248 \cdot 10^{-161} \lor \neg \left(y \le 2.06368582445499696 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x - y}{\frac{x \cdot x + y \cdot y}{y + x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 x 1.0) (< y 1.0))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2.0) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))