Average Error: 60.0 → 0.0
Time: 26.1s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{1}{9} + \left(x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{-1}{3}\right) \cdot \left(x \cdot \left(\frac{1}{45} \cdot x\right)\right)}{\frac{1}{91125} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{27}}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{1}{9} + \left(x \cdot \left(\frac{1}{45} \cdot x\right) + \frac{-1}{3}\right) \cdot \left(x \cdot \left(\frac{1}{45} \cdot x\right)\right)}{\frac{1}{91125} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{27}}}
double f(double x) {
        double r1889861 = 1.0;
        double r1889862 = x;
        double r1889863 = r1889861 / r1889862;
        double r1889864 = tan(r1889862);
        double r1889865 = r1889861 / r1889864;
        double r1889866 = r1889863 - r1889865;
        return r1889866;
}
↓
double f(double x) {
        double r1889867 = x;
        double r1889868 = 5.0;
        double r1889869 = pow(r1889867, r1889868);
        double r1889870 = 0.0021164021164021165;
        double r1889871 = r1889869 * r1889870;
        double r1889872 = 0.1111111111111111;
        double r1889873 = 0.022222222222222223;
        double r1889874 = r1889873 * r1889867;
        double r1889875 = r1889867 * r1889874;
        double r1889876 = -0.3333333333333333;
        double r1889877 = r1889875 + r1889876;
        double r1889878 = r1889877 * r1889875;
        double r1889879 = r1889872 + r1889878;
        double r1889880 = 1.0973936899862826e-05;
        double r1889881 = r1889867 * r1889867;
        double r1889882 = r1889867 * r1889881;
        double r1889883 = r1889882 * r1889882;
        double r1889884 = r1889880 * r1889883;
        double r1889885 = 0.037037037037037035;
        double r1889886 = r1889884 + r1889885;
        double r1889887 = r1889879 / r1889886;
        double r1889888 = r1889867 / r1889887;
        double r1889889 = r1889871 + r1889888;
        return r1889889;
}

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

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

  5. Applied flip3-+1.2

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

  6. Applied associate-*r/1.1

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

  7. Simplified0.3

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

  9. Applied associate-/l*0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 
(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))))