Average Error: 59.9 → 0.0
Time: 29.6s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{x}{\frac{1}{\frac{1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}} + {x}^{5} \cdot \frac{2}{945}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{x}{\frac{1}{\frac{1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)}} + {x}^{5} \cdot \frac{2}{945}
double f(double x) {
        double r2593731 = 1.0;
        double r2593732 = x;
        double r2593733 = r2593731 / r2593732;
        double r2593734 = tan(r2593732);
        double r2593735 = r2593731 / r2593734;
        double r2593736 = r2593733 - r2593735;
        return r2593736;
}
↓
double f(double x) {
        double r2593737 = x;
        double r2593738 = 1.0;
        double r2593739 = 0.3333333333333333;
        double r2593740 = 0.022222222222222223;
        double r2593741 = r2593737 * r2593737;
        double r2593742 = r2593740 * r2593741;
        double r2593743 = r2593739 + r2593742;
        double r2593744 = r2593738 / r2593743;
        double r2593745 = r2593737 / r2593744;
        double r2593746 = 5.0;
        double r2593747 = pow(r2593737, r2593746);
        double r2593748 = 0.0021164021164021165;
        double r2593749 = r2593747 * r2593748;
        double r2593750 = r2593745 + r2593749;
        return r2593750;
}

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

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

  5. Applied flip-+0.3

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

  6. Applied associate-*r/0.3

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

  8. Applied associate-/l*0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019146 
(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))))