Average Error: 19.8 → 5.5
Time: 2.1s
Precision: binary64
\[0 < x \land x < 1 \land y < 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.3361480154638368 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5102769436639434 \cdot 10^{-144} \lor \neg \left(y \leq 3.4054672957276274 \cdot 10^{-166}\right):\\ \;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1.3361480154638368 \cdot 10^{+154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \leq -1.5102769436639434 \cdot 10^{-144} \lor \neg \left(y \leq 3.4054672957276274 \cdot 10^{-166}\right):\\
\;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}}\right)\\

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

\end{array}
(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= y -1.3361480154638368e+154)
   -1.0
   (if (or (<= y -1.5102769436639434e-144)
           (not (<= y 3.4054672957276274e-166)))
     (log (exp (/ (- (* x x) (* y y)) (+ (* x x) (* y y)))))
     1.0)))
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 <= -1.3361480154638368e+154) {
		tmp = -1.0;
	} else if ((y <= -1.5102769436639434e-144) || !(y <= 3.4054672957276274e-166)) {
		tmp = log(exp(((x * x) - (y * y)) / ((x * x) + (y * y))));
	} else {
		tmp = 1.0;
	}
	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
Herbie5.5
\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.3361480154638368e154

    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 0

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

    if -1.3361480154638368e154 < y < -1.51027694366394339e-144 or 3.4054672957276274e-166 < y

    1. Initial program 0.2

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

    3. Applied add-log-exp_binary640.3

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

    4. Simplified0.3

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

    if -1.51027694366394339e-144 < y < 3.4054672957276274e-166

    1. Initial program 27.7

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

    2. Taylor expanded around inf 16.6

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3361480154638368 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5102769436639434 \cdot 10^{-144} \lor \neg \left(y \leq 3.4054672957276274 \cdot 10^{-166}\right):\\ \;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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))))