Average Error: 20.0 → 5.0
Time: 2.1s
Precision: binary64
\[0 < x \land x < 1 \land y < 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.3557348503118228 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5570222608799464 \cdot 10^{-162} \lor \neg \left(y \leq 1.5574783314889873 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1.3557348503118228 \cdot 10^{+154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \leq -1.5570222608799464 \cdot 10^{-162} \lor \neg \left(y \leq 1.5574783314889873 \cdot 10^{-162}\right):\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{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 -1.3557348503118228e+154)
   -1.0
   (if (or (<= y -1.5570222608799464e-162)
           (not (<= y 1.5574783314889873e-162)))
     (/ (* (- x y) (+ y x)) (+ (* 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 <= -1.3557348503118228e+154)) {
		tmp = -1.0;
	} else {
		double tmp_1;
		if (((y <= -1.5570222608799464e-162) || !(y <= 1.5574783314889873e-162))) {
			tmp_1 = (((double) (((double) (x - y)) * ((double) (y + x)))) / ((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

Original20.0
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 < -1.35573485031182282e154

    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 -1.35573485031182282e154 < y < -1.557022260879946e-162 or 1.55747833148898729e-162 < y

    1. Initial program Error: 0.0 bits

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

    if -1.557022260879946e-162 < y < 1.55747833148898729e-162

    1. Initial program Error: 29.9 bits

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

    2. Taylor expanded around inf Error: 15.6 bits

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

  4. Final simplificationError: 5.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3557348503118228 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5570222608799464 \cdot 10^{-162} \lor \neg \left(y \leq 1.5574783314889873 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))