Error

Target

Derivation

1. Initial program 59.9

   \[\frac{1}{x} - \frac{1}{\tan x}\]

2. Taylor expanded around 0 0.3

   \[\leadsto \color{blue}{\frac{1}{3} \cdot x + \left(\frac{1}{45} \cdot {x}^{3} + \frac{2}{945} \cdot {x}^{5}\right)}\]

3. Simplified0.3

   \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)\right) + {x}^{5} \cdot \frac{2}{945}}\]
4. Using strategy rm

5. Applied flip-+0.3

   \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{3} \cdot \frac{1}{3} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}{\frac{1}{3} - \frac{1}{45} \cdot \left(x \cdot x\right)}} + {x}^{5} \cdot \frac{2}{945}\]

6. Applied associate-*r/0.3

   \[\leadsto \color{blue}{\frac{x \cdot \left(\frac{1}{3} \cdot \frac{1}{3} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)\right)}{\frac{1}{3} - \frac{1}{45} \cdot \left(x \cdot x\right)}} + {x}^{5} \cdot \frac{2}{945}\]
7. Using strategy rm

8. Applied associate-/l*0.0

   \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{3} - \frac{1}{45} \cdot \left(x \cdot x\right)}{\frac{1}{3} \cdot \frac{1}{3} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}}} + {x}^{5} \cdot \frac{2}{945}\]

9. Final simplification0.0

   \[\leadsto \frac{x}{\frac{\frac{1}{3} - \frac{1}{45} \cdot \left(x \cdot x\right)}{\frac{1}{9} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}} + \frac{2}{945} \cdot {x}^{5}\]

Reproduce

herbie shell --seed 2019091 
(FPCore (x)
  :name "invcot (example 3.9)"
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

  (- (/ 1 x) (/ 1 (tan x))))