Average Error: 20.0 → 0.0
Time: 10.5s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\frac{-\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{-\mathsf{hypot}\left(x, y\right)}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\frac{-\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{-\mathsf{hypot}\left(x, y\right)}
double f(double x, double y) {
        double r118383 = x;
        double r118384 = y;
        double r118385 = r118383 - r118384;
        double r118386 = r118383 + r118384;
        double r118387 = r118385 * r118386;
        double r118388 = r118383 * r118383;
        double r118389 = r118384 * r118384;
        double r118390 = r118388 + r118389;
        double r118391 = r118387 / r118390;
        return r118391;
}
↓
double f(double x, double y) {
        double r118392 = x;
        double r118393 = y;
        double r118394 = r118392 - r118393;
        double r118395 = hypot(r118392, r118393);
        double r118396 = r118392 + r118393;
        double r118397 = r118395 / r118396;
        double r118398 = r118394 / r118397;
        double r118399 = -r118398;
        double r118400 = -r118395;
        double r118401 = r118399 / r118400;
        return r118401;
}

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
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Initial program 20.0

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

  3. Applied add-sqr-sqrt20.0

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

  4. Applied associate-/r*20.0

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

  5. Simplified20.0

    \[\leadsto \frac{\color{blue}{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}}{\sqrt{x \cdot x + y \cdot y}}\]
  6. Using strategy rm

  7. Applied frac-2neg20.0

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

  8. Simplified0.0

    \[\leadsto \frac{-\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{\color{blue}{-\mathsf{hypot}\left(x, y\right)}}\]

  9. Final simplification0.0

    \[\leadsto \frac{-\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{-\mathsf{hypot}\left(x, y\right)}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 x 1) (< y 1))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))