Average Error: 59.9 → 0.0
Time: 23.9s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{\left(x \cdot x\right) \cdot \frac{1}{45} + \frac{-1}{3}}{\left(x \cdot x\right) \cdot \frac{1}{45} + \frac{1}{3}}}{\left(x \cdot x\right) \cdot \frac{1}{45} + \frac{-1}{3}}}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
{x}^{5} \cdot \frac{2}{945} + \frac{x}{\frac{\frac{\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}{3}}}
double f(double x) {
        double r1581528 = 1.0;
        double r1581529 = x;
        double r1581530 = r1581528 / r1581529;
        double r1581531 = tan(r1581529);
        double r1581532 = r1581528 / r1581531;
        double r1581533 = r1581530 - r1581532;
        return r1581533;
}
↓
double f(double x) {
        double r1581534 = x;
        double r1581535 = 5.0;
        double r1581536 = pow(r1581534, r1581535);
        double r1581537 = 0.0021164021164021165;
        double r1581538 = r1581536 * r1581537;
        double r1581539 = r1581534 * r1581534;
        double r1581540 = 0.022222222222222223;
        double r1581541 = r1581539 * r1581540;
        double r1581542 = -0.3333333333333333;
        double r1581543 = r1581541 + r1581542;
        double r1581544 = 0.3333333333333333;
        double r1581545 = r1581541 + r1581544;
        double r1581546 = r1581543 / r1581545;
        double r1581547 = r1581546 / r1581543;
        double r1581548 = r1581534 / r1581547;
        double r1581549 = r1581538 + r1581548;
        return r1581549;
}

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

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

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 
(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))))