Error

Try it out

Target

Derivation

1. Split input into 3 regimes

2. if y < -1.3266524044529779e+154 or 1.8045349817621806e-222 < y < 2.0891795127861158e-165

   1. Initial program 55.1

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

   2. Taylor expanded around 0 10.2

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

   if -1.3266524044529779e+154 < y < -6.45209558740925e-161 or 2.0891795127861158e-165 < y

   1. Initial program 0.4

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

   if -6.45209558740925e-161 < y < 1.8045349817621806e-222

   1. Initial program 28.4

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

   3. Applied associate-/l*28.9

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

   4. Taylor expanded around inf 13.4

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

4. Final simplification6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3266524044529779 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -6.45209558740925 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\ \mathbf{elif}\;y \le 1.8045349817621806 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.0891795127861158 \cdot 10^{-165}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(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: 19.8s)Debug log