Average Error: 59.8 → 0.0
Time: 26.6s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{\frac{-1}{3} + \left(x \cdot x\right) \cdot \frac{1}{45}}{\frac{1}{3} + \left(x \cdot x\right) \cdot \frac{1}{45}}}{\frac{-1}{3} + \left(x \cdot x\right) \cdot \frac{1}{45}}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{x}{\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}}}
double f(double x) {
        double r3431745 = 1.0;
        double r3431746 = x;
        double r3431747 = r3431745 / r3431746;
        double r3431748 = tan(r3431746);
        double r3431749 = r3431745 / r3431748;
        double r3431750 = r3431747 - r3431749;
        return r3431750;
}
↓
double f(double x) {
        double r3431751 = x;
        double r3431752 = 5.0;
        double r3431753 = pow(r3431751, r3431752);
        double r3431754 = 0.0021164021164021165;
        double r3431755 = r3431753 * r3431754;
        double r3431756 = -0.3333333333333333;
        double r3431757 = r3431751 * r3431751;
        double r3431758 = 0.022222222222222223;
        double r3431759 = r3431757 * r3431758;
        double r3431760 = r3431756 + r3431759;
        double r3431761 = 0.3333333333333333;
        double r3431762 = r3431761 + r3431759;
        double r3431763 = r3431760 / r3431762;
        double r3431764 = r3431763 / r3431760;
        double r3431765 = r3431751 / r3431764;
        double r3431766 = r3431755 + r3431765;
        return r3431766;
}

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

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 
(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))))