Average Error: 59.8 → 0.2
Time: 28.0s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{91125}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{1}{27}\right) \cdot x}{\left(\frac{1}{9} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3}\right) + \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{91125}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{1}{27}\right) \cdot x}{\left(\frac{1}{9} - \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3}\right) + \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}
double f(double x) {
        double r3578537 = 1.0;
        double r3578538 = x;
        double r3578539 = r3578537 / r3578538;
        double r3578540 = tan(r3578538);
        double r3578541 = r3578537 / r3578540;
        double r3578542 = r3578539 - r3578541;
        return r3578542;
}
↓
double f(double x) {
        double r3578543 = x;
        double r3578544 = 5.0;
        double r3578545 = pow(r3578543, r3578544);
        double r3578546 = 0.0021164021164021165;
        double r3578547 = r3578545 * r3578546;
        double r3578548 = r3578543 * r3578543;
        double r3578549 = 1.0973936899862826e-05;
        double r3578550 = r3578548 * r3578549;
        double r3578551 = r3578548 * r3578548;
        double r3578552 = r3578550 * r3578551;
        double r3578553 = 0.037037037037037035;
        double r3578554 = r3578552 + r3578553;
        double r3578555 = r3578554 * r3578543;
        double r3578556 = 0.1111111111111111;
        double r3578557 = 0.022222222222222223;
        double r3578558 = r3578557 * r3578548;
        double r3578559 = 0.3333333333333333;
        double r3578560 = r3578558 * r3578559;
        double r3578561 = r3578556 - r3578560;
        double r3578562 = r3578558 * r3578558;
        double r3578563 = r3578561 + r3578562;
        double r3578564 = r3578555 / r3578563;
        double r3578565 = r3578547 + r3578564;
        return r3578565;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.8
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

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 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. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019168 
(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))))