Average Error: 20.2 → 0.0
Time: 10.7s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}
double f(double x, double y) {
        double r87322 = x;
        double r87323 = y;
        double r87324 = r87322 - r87323;
        double r87325 = r87322 + r87323;
        double r87326 = r87324 * r87325;
        double r87327 = r87322 * r87322;
        double r87328 = r87323 * r87323;
        double r87329 = r87327 + r87328;
        double r87330 = r87326 / r87329;
        return r87330;
}
↓
double f(double x, double y) {
        double r87331 = x;
        double r87332 = y;
        double r87333 = r87331 - r87332;
        double r87334 = hypot(r87331, r87332);
        double r87335 = r87333 / r87334;
        double r87336 = r87331 + r87332;
        double r87337 = r87334 / r87336;
        double r87338 = r87335 / r87337;
        return r87338;
}

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
Target0.1
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 20.2

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

  3. Applied add-sqr-sqrt20.2

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

  4. Applied associate-/r*20.3

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

  5. Simplified20.3

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

  7. Applied associate-/r/20.3

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

  8. Applied associate-/l*20.3

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 +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))))