Error

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Target

Derivation

1. Split input into 3 regimes

2. if y < -2.2820246712762874e-41

   1. Initial program 27.1

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

   3. Applied *-un-lft-identity27.1

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

   4. Applied times-frac26.6

      \[\leadsto \color{blue}{\frac{x - y}{1} \cdot \frac{x + y}{x \cdot x + y \cdot y}}\]

   5. Simplified26.6

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

   6. Taylor expanded around 0 0.5

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

   if -2.2820246712762874e-41 < y < -7.402706151578305e-166 or 3.611390805282423e-195 < y

   1. Initial program 3.4

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

   3. Applied *-un-lft-identity3.4

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

   4. Applied times-frac4.3

      \[\leadsto \color{blue}{\frac{x - y}{1} \cdot \frac{x + y}{x \cdot x + y \cdot y}}\]

   5. Simplified4.3

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

   if -7.402706151578305e-166 < y < 3.611390805282423e-195

   1. Initial program 30.3

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

   3. Applied clear-num30.3

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

   4. Taylor expanded around -inf 13.3

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

4. Final simplification5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.2820246712762874 \cdot 10^{-41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -7.402706151578305 \cdot 10^{-166} \lor \neg \left(y \le 3.611390805282423 \cdot 10^{-195}\right):\\ \;\;\;\;\frac{y + x}{y \cdot y + x \cdot x} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(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: 18.1s)Debug log