Average Error: 59.9 → 0.0
Time: 38.3s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{x}{\frac{\frac{1}{9} + \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)\right)}{\frac{1}{91125} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) + \frac{1}{27}}} + {x}^{5} \cdot \frac{2}{945}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{x}{\frac{\frac{1}{9} + \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)\right)}{\frac{1}{91125} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) + \frac{1}{27}}} + {x}^{5} \cdot \frac{2}{945}
double f(double x) {
        double r2773268 = 1.0;
        double r2773269 = x;
        double r2773270 = r2773268 / r2773269;
        double r2773271 = tan(r2773269);
        double r2773272 = r2773268 / r2773271;
        double r2773273 = r2773270 - r2773272;
        return r2773273;
}
↓
double f(double x) {
        double r2773274 = x;
        double r2773275 = 0.1111111111111111;
        double r2773276 = 0.022222222222222223;
        double r2773277 = r2773274 * r2773274;
        double r2773278 = r2773276 * r2773277;
        double r2773279 = -0.3333333333333333;
        double r2773280 = r2773279 + r2773278;
        double r2773281 = r2773278 * r2773280;
        double r2773282 = r2773275 + r2773281;
        double r2773283 = 1.0973936899862826e-05;
        double r2773284 = r2773277 * r2773277;
        double r2773285 = r2773284 * r2773277;
        double r2773286 = r2773283 * r2773285;
        double r2773287 = 0.037037037037037035;
        double r2773288 = r2773286 + r2773287;
        double r2773289 = r2773282 / r2773288;
        double r2773290 = r2773274 / r2773289;
        double r2773291 = 5.0;
        double r2773292 = pow(r2773274, r2773291);
        double r2773293 = 0.0021164021164021165;
        double r2773294 = r2773292 * r2773293;
        double r2773295 = r2773290 + r2773294;
        return r2773295;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.9
Target0.1
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

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. Simplified0.3

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

  5. Applied flip3-+1.3

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

  6. Applied associate-*l/1.1

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

  7. Simplified0.3

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \frac{\color{blue}{x \cdot \left(\frac{1}{91125} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) + \frac{1}{27}\right)}}{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) - \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right)\right)}\]
  8. Using strategy rm

  9. Applied associate-/l*0.0

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \color{blue}{\frac{x}{\frac{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right) - \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{45}\right)\right)}{\frac{1}{91125} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) + \frac{1}{27}}}}\]

  10. Simplified0.0

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \frac{x}{\color{blue}{\frac{\frac{1}{9} + \left(\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}{\frac{1}{27} + \frac{1}{91125} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}}}\]

  11. Final simplification0.0

    \[\leadsto \frac{x}{\frac{\frac{1}{9} + \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)\right)}{\frac{1}{91125} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) + \frac{1}{27}}} + {x}^{5} \cdot \frac{2}{945}\]

Reproduce

herbie shell --seed 2019139 
(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))))