Average Error: 20.1 → 5.0
Time: 3.1s
Precision: binary64
\[0 < x \land x < 1 \land y < 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.3513757347195584 \cdot 10^{+154}:\\ \;\;\;\;-1 + 2 \cdot {\left(\frac{x}{y}\right)}^{2}\\ \mathbf{elif}\;y \leq -9.049599147467105 \cdot 10^{-163}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq -1.0294507382078244 \cdot 10^{-172}:\\ \;\;\;\;-1 + 2 \cdot {\left(\frac{x}{y}\right)}^{2}\\ \mathbf{elif}\;y \leq 2.2931306999859318 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1.3513757347195584 \cdot 10^{+154}:\\
\;\;\;\;-1 + 2 \cdot {\left(\frac{x}{y}\right)}^{2}\\

\mathbf{elif}\;y \leq -9.049599147467105 \cdot 10^{-163}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \leq -1.0294507382078244 \cdot 10^{-172}:\\
\;\;\;\;-1 + 2 \cdot {\left(\frac{x}{y}\right)}^{2}\\

\mathbf{elif}\;y \leq 2.2931306999859318 \cdot 10^{-169}:\\
\;\;\;\;1\\

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

\end{array}
(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= y -1.3513757347195584e+154)
   (+ -1.0 (* 2.0 (pow (/ x y) 2.0)))
   (if (<= y -9.049599147467105e-163)
     (/ (* (- x y) (+ y x)) (+ (* x x) (* y y)))
     (if (<= y -1.0294507382078244e-172)
       (+ -1.0 (* 2.0 (pow (/ x y) 2.0)))
       (if (<= y 2.2931306999859318e-169)
         1.0
         (/ (* (- x y) (+ y x)) (+ (* x x) (* y 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 (y <= -1.3513757347195584e+154) {
		tmp = -1.0 + (2.0 * pow((x / y), 2.0));
	} else if (y <= -9.049599147467105e-163) {
		tmp = ((x - y) * (y + x)) / ((x * x) + (y * y));
	} else if (y <= -1.0294507382078244e-172) {
		tmp = -1.0 + (2.0 * pow((x / y), 2.0));
	} else if (y <= 2.2931306999859318e-169) {
		tmp = 1.0;
	} else {
		tmp = ((x - y) * (y + x)) / ((x * x) + (y * 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

Original20.1
Target0.1
Herbie5.0
\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.35137573471955844e154 or -9.049599147467105e-163 < y < -1.02945073820782443e-172

    1. Initial program 62.2

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

    2. Taylor expanded around 0 4.1

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

    3. Simplified4.1

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

    4. Taylor expanded around 0 4.1

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

    5. Simplified1.8

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

    if -1.35137573471955844e154 < y < -9.049599147467105e-163 or 2.2931306999859318e-169 < y

    1. Initial program 0.4

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

    if -1.02945073820782443e-172 < y < 2.2931306999859318e-169

    1. Initial program 30.4

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

    2. Taylor expanded around inf 15.0

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

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3513757347195584 \cdot 10^{+154}:\\ \;\;\;\;-1 + 2 \cdot {\left(\frac{x}{y}\right)}^{2}\\ \mathbf{elif}\;y \leq -9.049599147467105 \cdot 10^{-163}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq -1.0294507382078244 \cdot 10^{-172}:\\ \;\;\;\;-1 + 2 \cdot {\left(\frac{x}{y}\right)}^{2}\\ \mathbf{elif}\;y \leq 2.2931306999859318 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))