Error

Target

Derivation

1. Initial program 60.0

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

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

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

5. Applied add-log-exp0.4

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

7. Applied add-cube-cbrt0.4

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

8. Applied log-prod0.4

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

9. Final simplification0.4

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

Reproduce

herbie shell --seed 2018360 +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: 11.9s)Debug log