Error

Target

Derivation

1. Initial program 59.9

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

2. Taylor expanded around 0 0.3

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

3. Applied simplify0.3

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

5. Applied add-cbrt-cube0.3

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

6. Applied simplify1.2

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

7. Taylor expanded around inf 47.8

   \[\leadsto \color{blue}{15 \cdot \frac{e^{\log \frac{1}{45} - 2 \cdot \log \left(\frac{1}{x}\right)}}{x} + \left(\frac{2}{945} \cdot {x}^{5} + x \cdot e^{\log \frac{1}{45} - 2 \cdot \log \left(\frac{1}{x}\right)}\right)}\]

8. Applied simplify0.3

   \[\leadsto \color{blue}{(\left(x \cdot \frac{1}{45}\right) \cdot 15 + \left((\frac{2}{945} \cdot \left({x}^{5}\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{45}\right)\right))_*\right))_*}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2020178 +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))))