invcot (example 3.9)

Average Error: 60.0 → 0.0

Time: 23.4s

Precision: 64

\[-0.026 \lt x \land x \lt 0.026\]

Math

\[\frac{1}{x} - \frac{1}{\tan x}\]

↓

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

C

double f(double x) {
        double r1430866 = 1.0;
        double r1430867 = x;
        double r1430868 = r1430866 / r1430867;
        double r1430869 = tan(r1430867);
        double r1430870 = r1430866 / r1430869;
        double r1430871 = r1430868 - r1430870;
        return r1430871;
}

↓

double f(double x) {
        double r1430872 = x;
        double r1430873 = 5.0;
        double r1430874 = pow(r1430872, r1430873);
        double r1430875 = 0.0021164021164021165;
        double r1430876 = r1430874 * r1430875;
        double r1430877 = 0.022222222222222223;
        double r1430878 = r1430872 * r1430872;
        double r1430879 = r1430877 * r1430878;
        double r1430880 = 0.3333333333333333;
        double r1430881 = r1430879 - r1430880;
        double r1430882 = exp(r1430879);
        double r1430883 = log(r1430882);
        double r1430884 = r1430881 * r1430883;
        double r1430885 = 0.1111111111111111;
        double r1430886 = r1430884 + r1430885;
        double r1430887 = 1.0973936899862826e-05;
        double r1430888 = r1430878 * r1430872;
        double r1430889 = r1430888 * r1430888;
        double r1430890 = r1430887 * r1430889;
        double r1430891 = 0.037037037037037035;
        double r1430892 = r1430890 + r1430891;
        double r1430893 = r1430886 / r1430892;
        double r1430894 = r1430872 / r1430893;
        double r1430895 = r1430876 + r1430894;
        return r1430895;
}

Error

Bits error versus x

Target

Original	60.0
Target	0.1
Herbie	0.0

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

Derivation

1. Initial program 60.0

   \[\frac{1}{x} - \frac{1}{\tan x}\]

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. Simplified 0.3

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

5. Applied flip3-+ 1.2

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

6. Applied associate-*r/ 1.1

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

7. Simplified 0.3

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

9. Applied associate-/l* 0.0

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

10. Simplified 0.0

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \frac{x}{\color{blue}{\frac{\left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right) - \frac{1}{3}\right) + \frac{1}{9}}{\frac{1}{27} + \frac{1}{91125} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}}\]
11. Using strategy rm

12. Applied add-log-exp 0.0

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

13. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019128 
(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))))