Average Error: 20.9 → 7.3
Time: 2.7s
Precision: binary64
\[0 < x \land x < 1 \land y < 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.360803354623427 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -5.983925060339704 \cdot 10^{-156}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq 6.780097591949427 \cdot 10^{-255}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.8489230072387054 \cdot 10^{-193}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1.360803354623427 \cdot 10^{+154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \leq -5.983925060339704 \cdot 10^{-156}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \leq 6.780097591949427 \cdot 10^{-255}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \leq 1.8489230072387054 \cdot 10^{-193}:\\
\;\;\;\;-1\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot x - y \cdot y}{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 -1.360803354623427e+154)
   -1.0
   (if (<= y -5.983925060339704e-156)
     (/ (* (- x y) (+ y x)) (+ (* x x) (* y y)))
     (if (<= y 6.780097591949427e-255)
       1.0
       (if (<= y 1.8489230072387054e-193)
         -1.0
         (/ (- (* x x) (* y y)) (+ (* 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 <= -1.360803354623427e+154) {
		tmp = -1.0;
	} else if (y <= -5.983925060339704e-156) {
		tmp = ((x - y) * (y + x)) / ((x * x) + (y * y));
	} else if (y <= 6.780097591949427e-255) {
		tmp = 1.0;
	} else if (y <= 1.8489230072387054e-193) {
		tmp = -1.0;
	} else {
		tmp = ((x * x) - (y * y)) / ((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

Original20.9
Target0.1
Herbie7.3
\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if y < -1.3608033546234269e154 or 6.78009759194942696e-255 < y < 1.84892300723870541e-193

    1. Initial program 54.3

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

    2. Taylor expanded around 0 13.2

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

    if -1.3608033546234269e154 < y < -5.9839250603397035e-156

    1. Initial program 0.0

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

    if -5.9839250603397035e-156 < y < 6.78009759194942696e-255

    1. Initial program 28.9

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

    2. Taylor expanded around inf 13.8

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

    if 1.84892300723870541e-193 < y

    1. Initial program 5.2

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

    3. Applied *-un-lft-identity_binary64_7675.2

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

    4. Applied associate-/r*_binary64_7115.2

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

    5. Simplified5.2

      \[\leadsto \frac{\color{blue}{x \cdot x - y \cdot y}}{x \cdot x + y \cdot y}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.360803354623427 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -5.983925060339704 \cdot 10^{-156}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq 6.780097591949427 \cdot 10^{-255}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.8489230072387054 \cdot 10^{-193}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020289 
(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))))