Average Error: 19.9 → 0.0
Time: 5.1s
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 r68386 = x;
        double r68387 = y;
        double r68388 = r68386 - r68387;
        double r68389 = r68386 + r68387;
        double r68390 = r68388 * r68389;
        double r68391 = r68386 * r68386;
        double r68392 = r68387 * r68387;
        double r68393 = r68391 + r68392;
        double r68394 = r68390 / r68393;
        return r68394;
}
↓
double f(double x, double y) {
        double r68395 = x;
        double r68396 = y;
        double r68397 = r68395 - r68396;
        double r68398 = hypot(r68395, r68396);
        double r68399 = r68397 / r68398;
        double r68400 = r68395 + r68396;
        double r68401 = r68398 / r68400;
        double r68402 = r68399 / r68401;
        return r68402;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.0
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 19.9

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

  3. Applied add-sqr-sqrt19.9

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

  4. Applied associate-/r*19.9

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

  5. Simplified19.9

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

  7. Applied hypot-def0.0

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

  9. Applied associate-/r/0.0

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

  10. Applied associate-/l*0.0

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

  11. 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 2020003 +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))))