Error

Target

Derivation

1. Initial program 59.9

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

2. Initial simplification59.9

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

3. Taylor expanded around 0 0.3

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

4. Simplified0.3

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

6. Applied add-sqr-sqrt0.3

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

8. Applied add-cube-cbrt0.4

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

9. Applied associate-*r*0.4

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

10. Taylor expanded around 0 0.3

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

11. Simplified0.3

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

12. Final simplification0.3

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

Runtime

Time bar (total: 37.3s)Debug logProfile

herbie shell --seed 2018351 +o rules:numerics
(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))))