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Derivation

1. Initial program 20.4

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

2. Initial simplification20.4

   \[\leadsto \frac{\left(x - y\right) \cdot \left(y + x\right)}{(x \cdot x + \left(y \cdot y\right))_*}\]
3. Using strategy rm

4. Applied add-sqr-sqrt20.4

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

5. Applied times-frac20.4

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

6. Simplified20.4

   \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x^2 + y^2}^*}} \cdot \frac{y + x}{\sqrt{(x \cdot x + \left(y \cdot y\right))_*}}\]

7. Simplified0.0

   \[\leadsto \frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \color{blue}{\frac{x + y}{\sqrt{x^2 + y^2}^*}}\]
8. Using strategy rm

9. Applied *-un-lft-identity0.0

   \[\leadsto \frac{\color{blue}{1 \cdot \left(x - y\right)}}{\sqrt{x^2 + y^2}^*} \cdot \frac{x + y}{\sqrt{x^2 + y^2}^*}\]

10. Applied associate-/l*0.0

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{x^2 + y^2}^*}{x - y}}} \cdot \frac{x + y}{\sqrt{x^2 + y^2}^*}\]

11. Final simplification0.0

    \[\leadsto \frac{1}{\frac{\sqrt{x^2 + y^2}^*}{x - y}} \cdot \frac{x + y}{\sqrt{x^2 + y^2}^*}\]

Runtime

Time bar (total: 21.4s)Debug logProfile

herbie shell --seed 2018355 +o rules:numerics
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 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))))