Average Error: 19.9 → 0.0
Time: 23.2s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}
double f(double x, double y) {
        double r72640 = x;
        double r72641 = y;
        double r72642 = r72640 - r72641;
        double r72643 = r72640 + r72641;
        double r72644 = r72642 * r72643;
        double r72645 = r72640 * r72640;
        double r72646 = r72641 * r72641;
        double r72647 = r72645 + r72646;
        double r72648 = r72644 / r72647;
        return r72648;
}
↓
double f(double x, double y) {
        double r72649 = x;
        double r72650 = y;
        double r72651 = r72649 - r72650;
        double r72652 = hypot(r72649, r72650);
        double r72653 = r72651 / r72652;
        double r72654 = 1.0;
        double r72655 = r72649 + r72650;
        double r72656 = r72652 / r72655;
        double r72657 = r72654 / r72656;
        double r72658 = r72653 * r72657;
        return r72658;
}

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
Target0.1
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 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 times-frac20.0

    \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]

  5. Simplified20.0

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

  6. Simplified0.0

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

  8. Applied clear-num0.0

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

  9. Final simplification0.0

    \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\]

Reproduce

herbie shell --seed 2019323 +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))))