Error

Target

Derivation

1. Initial program 59.8

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

2. Taylor expanded around 0 0.3

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

4. Applied add-cube-cbrt1.5

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

5. Taylor expanded around inf 34.6

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

6. Applied simplify1.5

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

7. Taylor expanded around -inf 62.9

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

8. Applied simplify0.4

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

Runtime

Time bar (total: 39.2s)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' 
(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))))