Average Error: 19.9 → 0.0
Time: 18.8s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\left(\frac{x}{\mathsf{hypot}\left(x, y\right)} - \frac{y}{\mathsf{hypot}\left(x, y\right)}\right) \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\left(\frac{x}{\mathsf{hypot}\left(x, y\right)} - \frac{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 r55804 = x;
        double r55805 = y;
        double r55806 = r55804 - r55805;
        double r55807 = r55804 + r55805;
        double r55808 = r55806 * r55807;
        double r55809 = r55804 * r55804;
        double r55810 = r55805 * r55805;
        double r55811 = r55809 + r55810;
        double r55812 = r55808 / r55811;
        return r55812;
}
↓
double f(double x, double y) {
        double r55813 = x;
        double r55814 = y;
        double r55815 = hypot(r55813, r55814);
        double r55816 = r55813 / r55815;
        double r55817 = r55814 / r55815;
        double r55818 = r55816 - r55817;
        double r55819 = r55813 + r55814;
        double r55820 = r55819 / r55815;
        double r55821 = r55818 * r55820;
        return r55821;
}

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 div-sub0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +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))))