Average Error: 20.0 → 5.4
Time: 2.3s
Precision: binary64
\[0 < x \land x < 1 \land y < 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -3.345932334055194 \cdot 10^{-39}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5986692674923848 \cdot 10^{-162} \lor \neg \left(y \leq 2.3252284660370693 \cdot 10^{-157}\right):\\ \;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -3.345932334055194 \cdot 10^{-39}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \leq -1.5986692674923848 \cdot 10^{-162} \lor \neg \left(y \leq 2.3252284660370693 \cdot 10^{-157}\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 -3.345932334055194e-39)
   -1.0
   (if (or (<= y -1.5986692674923848e-162)
           (not (<= y 2.3252284660370693e-157)))
     (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 <= -3.345932334055194e-39) {
		tmp = -1.0;
	} else if ((y <= -1.5986692674923848e-162) || !(y <= 2.3252284660370693e-157)) {
		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

Original20.0
Target0.0
Herbie5.4
\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -3.3459323340551937e-39

    1. Initial program 28.1

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

    2. Taylor expanded around 0 0.5

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

    if -3.3459323340551937e-39 < y < -1.598669267492385e-162 or 2.3252284660370693e-157 < y

    1. Initial program 0.1

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

    3. Applied add-log-exp_binary64_11500.1

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

    4. Simplified0.1

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

    if -1.598669267492385e-162 < y < 2.3252284660370693e-157

    1. Initial program 29.4

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

    2. Taylor expanded around inf 16.0

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

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.345932334055194 \cdot 10^{-39}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5986692674923848 \cdot 10^{-162} \lor \neg \left(y \leq 2.3252284660370693 \cdot 10^{-157}\right):\\ \;\;\;\;\log \left(e^{\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020280 
(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))))