Average Error: 60.0 → 0.3
Time: 22.8s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{\frac{1}{91125} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{1}{27}}{\frac{\left(\frac{1}{9} - \left(x \cdot x\right) \cdot \frac{1}{135}\right) + \frac{1}{2025} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{x}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{\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}}{\frac{\left(\frac{1}{9} - \left(x \cdot x\right) \cdot \frac{1}{135}\right) + \frac{1}{2025} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{x}}
double f(double x) {
        double r1174885 = 1.0;
        double r1174886 = x;
        double r1174887 = r1174885 / r1174886;
        double r1174888 = tan(r1174886);
        double r1174889 = r1174885 / r1174888;
        double r1174890 = r1174887 - r1174889;
        return r1174890;
}
↓
double f(double x) {
        double r1174891 = x;
        double r1174892 = 5.0;
        double r1174893 = pow(r1174891, r1174892);
        double r1174894 = 0.0021164021164021165;
        double r1174895 = r1174893 * r1174894;
        double r1174896 = 1.0973936899862826e-05;
        double r1174897 = r1174891 * r1174891;
        double r1174898 = r1174897 * r1174891;
        double r1174899 = r1174898 * r1174898;
        double r1174900 = r1174896 * r1174899;
        double r1174901 = 0.037037037037037035;
        double r1174902 = r1174900 + r1174901;
        double r1174903 = 0.1111111111111111;
        double r1174904 = 0.007407407407407408;
        double r1174905 = r1174897 * r1174904;
        double r1174906 = r1174903 - r1174905;
        double r1174907 = 0.0004938271604938272;
        double r1174908 = r1174897 * r1174897;
        double r1174909 = r1174907 * r1174908;
        double r1174910 = r1174906 + r1174909;
        double r1174911 = r1174910 / r1174891;
        double r1174912 = r1174902 / r1174911;
        double r1174913 = r1174895 + r1174912;
        return r1174913;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.0
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

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

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

  5. Applied flip3-+1.2

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

  6. Applied associate-*r/1.1

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

  7. Simplified0.3

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

  9. Applied associate-/l*0.3

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

  10. Simplified0.3

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \frac{\frac{1}{27} + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{91125}}{\color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{2025} + \left(\frac{1}{9} - \frac{1}{135} \cdot \left(x \cdot x\right)\right)}{x}}}\]

  11. Final simplification0.3

    \[\leadsto {x}^{5} \cdot \frac{2}{945} + \frac{\frac{1}{91125} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{1}{27}}{\frac{\left(\frac{1}{9} - \left(x \cdot x\right) \cdot \frac{1}{135}\right) + \frac{1}{2025} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{x}}\]

Reproduce

herbie shell --seed 2019156 
(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))))