Average Error: 20.1 → 0.0
Time: 4.3s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\frac{x}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}} - \frac{y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\frac{x}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}} - \frac{y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}
double f(double x, double y) {
        double r107542 = x;
        double r107543 = y;
        double r107544 = r107542 - r107543;
        double r107545 = r107542 + r107543;
        double r107546 = r107544 * r107545;
        double r107547 = r107542 * r107542;
        double r107548 = r107543 * r107543;
        double r107549 = r107547 + r107548;
        double r107550 = r107546 / r107549;
        return r107550;
}
↓
double f(double x, double y) {
        double r107551 = x;
        double r107552 = y;
        double r107553 = hypot(r107551, r107552);
        double r107554 = r107551 + r107552;
        double r107555 = r107553 / r107554;
        double r107556 = r107553 * r107555;
        double r107557 = r107551 / r107556;
        double r107558 = r107552 / r107556;
        double r107559 = r107557 - r107558;
        return r107559;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
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 20.1

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

  2. Simplified20.1

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

  4. Applied *-un-lft-identity20.1

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

  5. Applied add-sqr-sqrt20.1

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

  6. Applied times-frac20.1

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

  7. Simplified20.1

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

  8. Simplified0.0

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

  10. Applied div-sub0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020100 +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))))