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

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

5. Applied fma-udef0.3

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

6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019007 +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))))

Details

Time bar (total: 20.8s)Debug log