Average Error: 20.3 → 0.1
Time: 23.0s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\left(\left(\sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}} \cdot \sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}\right) \cdot \sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}\right) \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\left(\left(\sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}} \cdot \sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}\right) \cdot \sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}\right) \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}
double f(double x, double y) {
        double r68587 = x;
        double r68588 = y;
        double r68589 = r68587 - r68588;
        double r68590 = r68587 + r68588;
        double r68591 = r68589 * r68590;
        double r68592 = r68587 * r68587;
        double r68593 = r68588 * r68588;
        double r68594 = r68592 + r68593;
        double r68595 = r68591 / r68594;
        return r68595;
}
↓
double f(double x, double y) {
        double r68596 = x;
        double r68597 = y;
        double r68598 = r68596 - r68597;
        double r68599 = hypot(r68596, r68597);
        double r68600 = r68598 / r68599;
        double r68601 = cbrt(r68600);
        double r68602 = r68601 * r68601;
        double r68603 = r68602 * r68601;
        double r68604 = r68596 + r68597;
        double r68605 = r68604 / r68599;
        double r68606 = r68603 * r68605;
        return r68606;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.3
Target0.0
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Initial program 20.3

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

  3. Applied add-sqr-sqrt20.3

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

  4. Applied times-frac20.3

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

  5. Simplified20.3

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

  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 add-cube-cbrt0.1

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}} \cdot \sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}\right) \cdot \sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\]

  9. Final simplification0.1

    \[\leadsto \left(\left(\sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}} \cdot \sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}\right) \cdot \sqrt[3]{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}\right) \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\]

Reproduce

herbie shell --seed 2019306 +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))))