Average Error: 60.0 → 0.0
Time: 39.4s
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 r2805146 = 1.0;
        double r2805147 = x;
        double r2805148 = r2805146 / r2805147;
        double r2805149 = tan(r2805147);
        double r2805150 = r2805146 / r2805149;
        double r2805151 = r2805148 - r2805150;
        return r2805151;
}
↓
double f(double x) {
        double r2805152 = x;
        double r2805153 = 0.1111111111111111;
        double r2805154 = 0.022222222222222223;
        double r2805155 = r2805152 * r2805152;
        double r2805156 = r2805154 * r2805155;
        double r2805157 = -0.3333333333333333;
        double r2805158 = r2805157 + r2805156;
        double r2805159 = r2805156 * r2805158;
        double r2805160 = r2805153 + r2805159;
        double r2805161 = 1.0973936899862826e-05;
        double r2805162 = r2805155 * r2805155;
        double r2805163 = r2805162 * r2805155;
        double r2805164 = r2805161 * r2805163;
        double r2805165 = 0.037037037037037035;
        double r2805166 = r2805164 + r2805165;
        double r2805167 = r2805160 / r2805166;
        double r2805168 = r2805152 / r2805167;
        double r2805169 = 5.0;
        double r2805170 = pow(r2805152, r2805169);
        double r2805171 = 0.0021164021164021165;
        double r2805172 = r2805170 * r2805171;
        double r2805173 = r2805168 + r2805172;
        return r2805173;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.0
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 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

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

  5. Applied flip3-+1.2

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

  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 2019141 
(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))))