Average Error: 20.2 → 5.6
Time: 2.3s
Precision: binary64
\[0 < x \land x < 1 \land y < 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -6.71233735076048 \cdot 10^{-40}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5799804884119341 \cdot 10^{-162} \lor \neg \left(y \leq 4.755047945101056 \cdot 10^{-204}\right):\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \left(\left(y + x\right) \cdot \frac{1}{\sqrt{x \cdot x + y \cdot y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -6.71233735076048 \cdot 10^{-40}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \leq -1.5799804884119341 \cdot 10^{-162} \lor \neg \left(y \leq 4.755047945101056 \cdot 10^{-204}\right):\\
\;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \left(\left(y + x\right) \cdot \frac{1}{\sqrt{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 -6.71233735076048e-40)
   -1.0
   (if (or (<= y -1.5799804884119341e-162) (not (<= y 4.755047945101056e-204)))
     (*
      (/ (- x y) (sqrt (+ (* x x) (* y y))))
      (* (+ y x) (/ 1.0 (sqrt (+ (* 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 <= -6.71233735076048e-40)) {
		tmp = -1.0;
	} else {
		double tmp_1;
		if (((y <= -1.5799804884119341e-162) || !(y <= 4.755047945101056e-204))) {
			tmp_1 = ((double) ((((double) (x - y)) / ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))))) * ((double) (((double) (y + x)) * (1.0 / ((double) sqrt(((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

Original20.2
Target0.0
Herbie5.6
\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -6.71233735076047962e-40

    1. Initial program 28.5

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

    2. Taylor expanded around 0 0.6

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

    if -6.71233735076047962e-40 < y < -1.5799804884119341e-162 or 4.755047945101056e-204 < y

    1. Initial program 4.2

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

    3. Applied add-sqr-sqrt_binary644.2

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

    4. Applied times-frac_binary644.7

      \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]
    5. Using strategy rm

    6. Applied div-inv_binary644.9

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

    if -1.5799804884119341e-162 < y < 4.755047945101056e-204

    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 13.9

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.71233735076048 \cdot 10^{-40}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.5799804884119341 \cdot 10^{-162} \lor \neg \left(y \leq 4.755047945101056 \cdot 10^{-204}\right):\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \left(\left(y + x\right) \cdot \frac{1}{\sqrt{x \cdot x + y \cdot y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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))))