Error

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Target

Derivation

1. Split input into 3 regimes

2. if y < -1.3385669613614879e+154

   1. Initial program 63.6

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

   2. Taylor expanded around 0 0

      \[\leadsto \color{blue}{-1}\]

   if -1.3385669613614879e+154 < y < -4.11638679057694e-162 or 1.6449638335729248e-162 < y

   1. Initial program 0.1

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

   3. Applied associate-/l*0.5

      \[\leadsto \color{blue}{\frac{x - y}{\frac{x \cdot x + y \cdot y}{x + y}}}\]
   4. Using strategy rm

   5. Applied flip-+0.1

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

   6. Applied associate-/r/0.1

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

   7. Applied associate-/r*0.1

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

   9. Applied +-commutative0.1

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

   if -4.11638679057694e-162 < y < 1.6449638335729248e-162

   1. Initial program 29.6

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

   3. Applied associate-/l*30.0

      \[\leadsto \color{blue}{\frac{x - y}{\frac{x \cdot x + y \cdot y}{x + y}}}\]
   4. Using strategy rm

   5. Applied flip-+29.6

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

   6. Applied associate-/r/29.6

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

   7. Applied associate-/r*29.6

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

   8. Taylor expanded around -inf 15.2

      \[\leadsto \color{blue}{1}\]
3. Recombined 3 regimes into one program.

4. Final simplification4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3385669613614879 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -4.11638679057694 \cdot 10^{-162} \lor \neg \left(y \le 1.6449638335729248 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{\frac{x - y}{\frac{y \cdot y + x \cdot x}{x \cdot x - y \cdot y}}}{x - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(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))))

Details

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