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 0.4

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

4. Simplified0.4

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

6. Applied add-cbrt-cube0.4

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

7. Taylor expanded around 0 0.8

   \[\leadsto \color{blue}{\frac{1}{15} \cdot \left({x}^{3} \cdot {\frac{1}{27}}^{\frac{1}{3}}\right) + \left(\frac{2}{945} \cdot {x}^{5} + x \cdot {\frac{1}{27}}^{\frac{1}{3}}\right)}\]

8. Simplified0.4

   \[\leadsto \color{blue}{(\left(\sqrt[3]{\frac{1}{27}}\right) \cdot \left((\left(x \cdot x\right) \cdot \left(\frac{1}{15} \cdot x\right) + x)_*\right) + \left({x}^{5} \cdot \frac{2}{945}\right))_*}\]

9. Final simplification0.4

   \[\leadsto (\left(\sqrt[3]{\frac{1}{27}}\right) \cdot \left((\left(x \cdot x\right) \cdot \left(\frac{1}{15} \cdot x\right) + x)_*\right) + \left(\frac{2}{945} \cdot {x}^{5}\right))_*\]

Runtime

Time bar (total: 31.6s)Debug logProfile

herbie shell --seed 2018251 +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))))