Average Error: 20.4 → 0.0
Time: 2.3s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}
double code(double x, double y) {
	return ((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) (x - y)) / ((double) hypot(x, y)))) * ((double) (((double) (x + y)) / ((double) 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

Original20.4
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 20.4

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

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

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

  4. Applied times-frac20.7

    \[\leadsto \color{blue}{\frac{x - y}{1} \cdot \frac{x + y}{x \cdot x + y \cdot y}}\]

  5. Simplified20.7

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

  7. Applied add-sqr-sqrt20.7

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

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

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

  9. Applied times-frac20.6

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

  10. Simplified20.6

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

  11. Simplified0.2

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

  13. Applied associate-*r*0.2

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

  14. Simplified0.0

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

  15. Final simplification0.0

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

Reproduce

herbie shell --seed 2020121 +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))))