invcot (example 3.9)

Average Error: 59.9 → 0.3

Time: 30.7s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r2851277 = 1.0;
        double r2851278 = x;
        double r2851279 = r2851277 / r2851278;
        double r2851280 = tan(r2851278);
        double r2851281 = r2851277 / r2851280;
        double r2851282 = r2851279 - r2851281;
        return r2851282;
}

↓

double f(double x) {
        double r2851283 = x;
        double r2851284 = 5.0;
        double r2851285 = pow(r2851283, r2851284);
        double r2851286 = 0.0021164021164021165;
        double r2851287 = r2851285 * r2851286;
        double r2851288 = r2851283 * r2851283;
        double r2851289 = r2851288 * r2851283;
        double r2851290 = 1.0973936899862826e-05;
        double r2851291 = r2851289 * r2851290;
        double r2851292 = r2851291 * r2851289;
        double r2851293 = 0.037037037037037035;
        double r2851294 = r2851292 + r2851293;
        double r2851295 = 0.022222222222222223;
        double r2851296 = r2851283 * r2851295;
        double r2851297 = r2851283 * r2851296;
        double r2851298 = 0.3333333333333333;
        double r2851299 = r2851297 - r2851298;
        double r2851300 = r2851297 * r2851299;
        double r2851301 = 0.1111111111111111;
        double r2851302 = r2851300 + r2851301;
        double r2851303 = r2851283 / r2851302;
        double r2851304 = r2851294 * r2851303;
        double r2851305 = r2851287 + r2851304;
        return r2851305;
}

Error

Bits error versus x

Target

Original	59.9
Target	0.1
Herbie	0.3

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

Derivation

1. Initial program 59.9

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

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. Simplified 0.3

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

5. Applied flip3-+ 1.2

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

6. Applied associate-*r/ 1.1

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

7. Simplified 0.3

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

9. Applied *-un-lft-identity 0.3

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

10. Applied times-frac 0.3

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

11. Simplified 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019165 
(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))))