Error

Try it out

Target

Derivation

1. Split input into 3 regimes

2. if y < -1.3362591618789198e+154

   1. Initial program 63.6

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

   2. Initial simplification63.6

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

   3. Taylor expanded around 0 0

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

   if -1.3362591618789198e+154 < y < -1.2206913091651824e-159 or 1.0395458390998354e-187 < y

   1. Initial program 1.5

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

   2. Initial simplification1.5

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

   if -1.2206913091651824e-159 < y < 1.0395458390998354e-187

   1. Initial program 30.2

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

   2. Initial simplification30.2

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

   3. Taylor expanded around -inf 15.0

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

4. Final simplification5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3362591618789198 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.2206913091651824 \cdot 10^{-159} \lor \neg \left(y \le 1.0395458390998354 \cdot 10^{-187}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(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

Time bar (total: 12.5s)Debug log