Error

Try it out

Target

Derivation

1. Split input into 3 regimes

2. if y < -1.3665520024672976e+154 or 2.9020243613441586e-202 < y < 2.5202678793515666e-174

   1. Initial program 58.1

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

   2. Initial simplification58.1

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

   4. Applied associate-/l*56.8

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

   5. Taylor expanded around 0 6.1

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

   if -1.3665520024672976e+154 < y < -5.309205025385931e-161 or 2.5202678793515666e-174 < y

   1. Initial program 0.8

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

   2. Initial simplification0.8

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

   if -5.309205025385931e-161 < y < 2.9020243613441586e-202

   1. Initial program 29.5

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

   2. Initial simplification29.5

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

   4. Applied associate-/l*30.1

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

   5. Taylor expanded around -inf 14.1

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

4. Final simplification5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3665520024672976 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -5.309205025385931 \cdot 10^{-161}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\ \mathbf{elif}\;y \le 2.9020243613441586 \cdot 10^{-202}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 2.5202678793515666 \cdot 10^{-174}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 
(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: 11.1s)Debug log