Average Error: 20.0 → 0.0
Time: 5.7s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\frac{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{1 \cdot \mathsf{hypot}\left(x, y\right)}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\frac{\frac{x - y}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}{1 \cdot \mathsf{hypot}\left(x, y\right)}
double code(double x, double y) {
	return (((x - y) * (x + y)) / ((x * x) + (y * y)));
}
↓
double code(double x, double y) {
	return (((x - y) / (hypot(x, y) / (x + y))) / (1.0 * hypot(x, y)));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
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 20.0

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

  3. Applied add-sqr-sqrt20.0

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

  4. Applied associate-/r*20.0

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

  5. Simplified20.0

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

  7. Applied *-un-lft-identity20.0

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

  8. Applied sqrt-prod20.0

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

  9. Simplified20.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +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))))