Error

Target

Derivation

1. Split input into 3 regimes

2. if x < -0.9934881645172994

   1. Initial program 20.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied flip--50.0

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}}\]

   4. Applied associate-/r/52.5

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \color{blue}{\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)}\]

   5. Taylor expanded around -inf 0.9

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)}\]

   6. Simplified0.4

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}}\]
   7. Using strategy rm

   8. Applied div-inv0.4

      \[\leadsto \left(\color{blue}{\frac{2}{x} \cdot \frac{1}{x \cdot x}} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}\]

   9. Applied fma-def0.4

      \[\leadsto \color{blue}{(\left(\frac{2}{x}\right) \cdot \left(\frac{1}{x \cdot x}\right) + \left(\frac{2}{{x}^{7}}\right))_*} + \frac{2}{{x}^{5}}\]

   if -0.9934881645172994 < x < 129.39195686841717

   1. Initial program 0.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied flip--0.0

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}}\]

   4. Applied associate-/r/0.0

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \color{blue}{\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt0.0

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x \cdot x - 1 \cdot 1} \cdot \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}\]

   7. Applied associate-*r*0.0

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \color{blue}{\left(\frac{1}{x \cdot x - 1 \cdot 1} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1}}\]

   if 129.39195686841717 < x

   1. Initial program 20.1

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied flip--50.2

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}}\]

   4. Applied associate-/r/52.8

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \color{blue}{\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)}\]

   5. Taylor expanded around -inf 0.5

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)}\]

   6. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{5}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9934881645172994:\\ \;\;\;\;(\left(\frac{2}{x}\right) \cdot \left(\frac{1}{x \cdot x}\right) + \left(\frac{2}{{x}^{7}}\right))_* + \frac{2}{{x}^{5}}\\ \mathbf{elif}\;x \le 129.39195686841717:\\ \;\;\;\;\sqrt{1 + x} \cdot \left(\sqrt{1 + x} \cdot \frac{1}{x \cdot x - 1}\right) + \left(\frac{1}{1 + x} - \frac{2}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x}}{x \cdot x}\right) + \frac{2}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))

Details

Time bar (total: 1.2m)Debug log