Average Error: 19.8 → 4.7
Time: 6.2s
Precision: binary64
\[0 < x \land x < 1 \land y < 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 1:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot y}{x \cdot x - y \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 1:\\
\;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot y}{x \cdot x - y \cdot y}}\\

\mathbf{else}:\\
\;\;\;\;-1 + 2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\

\end{array}
(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) 1.0)
   (/ 1.0 (/ (+ (* x x) (* y y)) (- (* x x) (* y y))))
   (+ -1.0 (* 2.0 (* (/ x y) (/ x y))))))
double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}
↓
double code(double x, double y) {
	double tmp;
	if ((((x - y) * (x + y)) / ((x * x) + (y * y))) <= 1.0) {
		tmp = 1.0 / (((x * x) + (y * y)) / ((x * x) - (y * y)));
	} else {
		tmp = -1.0 + (2.0 * ((x / y) * (x / y)));
	}
	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

Original19.8
Target0.1
Herbie4.7
\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 1

    1. Initial program 0.0

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

    3. Applied clear-num_binary64_34870.0

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

    4. Simplified0.0

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

    if 1 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

    1. Initial program 64.0

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

    2. Taylor expanded around 0 31.2

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

    3. Simplified31.2

      \[\leadsto \color{blue}{-1 + 2 \cdot \frac{x \cdot x}{y \cdot y}}\]
    4. Using strategy rm

    5. Applied times-frac_binary64_349415.1

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

  4. Final simplification4.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 1:\\ \;\;\;\;\frac{1}{\frac{x \cdot x + y \cdot y}{x \cdot x - y \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021096 
(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))))