Average Error: 20.5 → 4.8
Time: 2.3s
Precision: binary64
\[0 < x \land x < 1 \land y < 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.3297178600730637 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5937452839923381 \cdot 10^{-161} \lor \neg \left(y \leq 1.9735231634376482 \cdot 10^{-169}\right):\\ \;\;\;\;\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1.3297178600730637 \cdot 10^{+154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \leq -1.5937452839923381 \cdot 10^{-161} \lor \neg \left(y \leq 1.9735231634376482 \cdot 10^{-169}\right):\\
\;\;\;\;\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\\

\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 -1.3297178600730637e+154)
   -1.0
   (if (or (<= y -1.5937452839923381e-161)
           (not (<= y 1.9735231634376482e-169)))
     (/ (- (* x x) (* y y)) (+ (* x x) (* y y)))
     1.0)))
double code(double x, double y) {
	return (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y)))));
}
↓
double code(double x, double y) {
	double tmp;
	if ((y <= -1.3297178600730637e+154)) {
		tmp = -1.0;
	} else {
		double tmp_1;
		if (((y <= -1.5937452839923381e-161) || !(y <= 1.9735231634376482e-169))) {
			tmp_1 = (((double) (((double) (x * x)) - ((double) (y * y)))) / ((double) (((double) (x * x)) + ((double) (y * y)))));
		} else {
			tmp_1 = 1.0;
		}
		tmp = tmp_1;
	}
	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

Original20.5
Target0.1
Herbie4.8
\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.3297178600730637e154

    1. Initial program Error: 64.0 bits

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

    2. Taylor expanded around 0 Error: 0 bits

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

    if -1.3297178600730637e154 < y < -1.5937452839923381e-161 or 1.9735231634376482e-169 < y

    1. Initial program Error: 0.3 bits

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

    3. Applied *-un-lft-identityError: 0.3 bits

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

    4. Applied associate-/r*Error: 0.3 bits

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

    5. SimplifiedError: 0.3 bits

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

    if -1.5937452839923381e-161 < y < 1.9735231634376482e-169

    1. Initial program Error: 30.7 bits

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

    2. Taylor expanded around inf Error: 14.9 bits

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

  4. Final simplificationError: 4.8 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3297178600730637 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5937452839923381 \cdot 10^{-161} \lor \neg \left(y \leq 1.9735231634376482 \cdot 10^{-169}\right):\\ \;\;\;\;\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))))