Average Error: 59.8 → 0.3
Time: 29.6s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{\left(\frac{1}{27} + \frac{\left(x \cdot x\right) \cdot x}{\frac{91125}{\left(x \cdot x\right) \cdot x}}\right) \cdot x}{\frac{1}{9} + \left(\frac{x \cdot x}{45} \cdot \frac{x \cdot x}{45} - \frac{1}{3} \cdot \frac{x \cdot x}{45}\right)}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{\left(\frac{1}{27} + \frac{\left(x \cdot x\right) \cdot x}{\frac{91125}{\left(x \cdot x\right) \cdot x}}\right) \cdot x}{\frac{1}{9} + \left(\frac{x \cdot x}{45} \cdot \frac{x \cdot x}{45} - \frac{1}{3} \cdot \frac{x \cdot x}{45}\right)}
double f(double x) {
        double r2571592 = 1.0;
        double r2571593 = x;
        double r2571594 = r2571592 / r2571593;
        double r2571595 = tan(r2571593);
        double r2571596 = r2571592 / r2571595;
        double r2571597 = r2571594 - r2571596;
        return r2571597;
}
↓
double f(double x) {
        double r2571598 = x;
        double r2571599 = 5.0;
        double r2571600 = pow(r2571598, r2571599);
        double r2571601 = 0.0021164021164021165;
        double r2571602 = r2571600 * r2571601;
        double r2571603 = 0.037037037037037035;
        double r2571604 = r2571598 * r2571598;
        double r2571605 = r2571604 * r2571598;
        double r2571606 = 91125.0;
        double r2571607 = r2571606 / r2571605;
        double r2571608 = r2571605 / r2571607;
        double r2571609 = r2571603 + r2571608;
        double r2571610 = r2571609 * r2571598;
        double r2571611 = 0.1111111111111111;
        double r2571612 = 45.0;
        double r2571613 = r2571604 / r2571612;
        double r2571614 = r2571613 * r2571613;
        double r2571615 = 0.3333333333333333;
        double r2571616 = r2571615 * r2571613;
        double r2571617 = r2571614 - r2571616;
        double r2571618 = r2571611 + r2571617;
        double r2571619 = r2571610 / r2571618;
        double r2571620 = r2571602 + r2571619;
        return r2571620;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 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

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

  5. Applied flip3-+1.2

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

  6. Applied associate-*r/1.1

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019129 
(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))))