Average Error: 59.9 → 0.0
Time: 45.7s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{x}{\frac{\frac{1}{3} - \frac{1}{45} \cdot \left(x \cdot x\right)}{\frac{1}{9} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}} + \frac{2}{945} \cdot {x}^{5}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{x}{\frac{\frac{1}{3} - \frac{1}{45} \cdot \left(x \cdot x\right)}{\frac{1}{9} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}} + \frac{2}{945} \cdot {x}^{5}
double f(double x) {
        double r6815929 = 1.0;
        double r6815930 = x;
        double r6815931 = r6815929 / r6815930;
        double r6815932 = tan(r6815930);
        double r6815933 = r6815929 / r6815932;
        double r6815934 = r6815931 - r6815933;
        return r6815934;
}
↓
double f(double x) {
        double r6815935 = x;
        double r6815936 = 0.3333333333333333;
        double r6815937 = 0.022222222222222223;
        double r6815938 = r6815935 * r6815935;
        double r6815939 = r6815937 * r6815938;
        double r6815940 = r6815936 - r6815939;
        double r6815941 = 0.1111111111111111;
        double r6815942 = r6815939 * r6815939;
        double r6815943 = r6815941 - r6815942;
        double r6815944 = r6815940 / r6815943;
        double r6815945 = r6815935 / r6815944;
        double r6815946 = 0.0021164021164021165;
        double r6815947 = 5.0;
        double r6815948 = pow(r6815935, r6815947);
        double r6815949 = r6815946 * r6815948;
        double r6815950 = r6815945 + r6815949;
        return r6815950;
}

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

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

  5. Applied flip-+0.3

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

  6. Applied associate-*r/0.3

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

  8. Applied associate-/l*0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019120 
(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))))