Error

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Target

Derivation

1. Split input into 3 regimes

2. if y < -1.3396022331219297e+154 or -1.0705598109984798e-175 < y < -2.4722449832826208e-216

   1. Initial program 56.9

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

   3. Applied add-sqr-sqrt56.9

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

   4. Applied associate-/r*56.9

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

   5. Taylor expanded around 0 8.1

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

   if -1.3396022331219297e+154 < y < -1.0705598109984798e-175 or 1.7274153437829284e-162 < y

   1. Initial program 0.7

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

   if -2.4722449832826208e-216 < y < 1.7274153437829284e-162

   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 add-sqr-sqrt28.4

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

   4. Applied associate-/r*28.4

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

   5. Taylor expanded around -inf 12.7

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

4. Final simplification5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.3396022331219297 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.0705598109984798 \cdot 10^{-175}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\ \mathbf{elif}\;y \le -2.4722449832826208 \cdot 10^{-216}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 1.7274153437829284 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(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: 26.5s)Debug log