Average Error: 59.9 → 0.5
Time: 29.3s
Precision: 64
Internal Precision: 2368
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\left(\frac{2}{945} \cdot {x}^{5} + \left(\left(\log \left(\sqrt{\sqrt{e^{\frac{1}{45} \cdot {x}^{3}}}}\right) + \log \left(\sqrt{\sqrt{e^{\frac{1}{45} \cdot {x}^{3}}}}\right)\right) + \log \left(\sqrt{e^{\frac{1}{45} \cdot {x}^{3}}}\right)\right)\right) + \frac{1}{3} \cdot x\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.9
Target0.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

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. Using strategy rm

  4. Applied add-log-exp0.5

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied log-prod0.5

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

  9. Applied add-sqr-sqrt0.5

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

  10. Applied sqrt-prod0.5

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

  11. Applied log-prod0.5

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

  12. Final simplification0.5

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

Runtime

Time bar (total: 29.3s)Debug logProfile

herbie shell --seed 2018220 
(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))))