Average Error: 59.8 → 0.0
Time: 23.5s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}{\frac{1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}}{\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}{\frac{1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}}{\frac{-1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}}
double f(double x) {
        double r2847361 = 1.0;
        double r2847362 = x;
        double r2847363 = r2847361 / r2847362;
        double r2847364 = tan(r2847362);
        double r2847365 = r2847361 / r2847364;
        double r2847366 = r2847363 - r2847365;
        return r2847366;
}
↓
double f(double x) {
        double r2847367 = x;
        double r2847368 = 5.0;
        double r2847369 = pow(r2847367, r2847368);
        double r2847370 = 0.0021164021164021165;
        double r2847371 = r2847369 * r2847370;
        double r2847372 = -0.3333333333333333;
        double r2847373 = 0.022222222222222223;
        double r2847374 = r2847367 * r2847367;
        double r2847375 = r2847373 * r2847374;
        double r2847376 = r2847372 + r2847375;
        double r2847377 = 0.3333333333333333;
        double r2847378 = r2847377 + r2847375;
        double r2847379 = r2847376 / r2847378;
        double r2847380 = r2847379 / r2847376;
        double r2847381 = r2847367 / r2847380;
        double r2847382 = r2847371 + r2847381;
        return r2847382;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.8
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 59.8

    \[\frac{1}{x} - \frac{1}{\tan x}\]

  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 flip-+0.3

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

  6. Applied associate-*r/0.3

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

  8. Applied associate-/l*0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 
(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))))