invcot (example 3.9)

Average Error: 59.8 → 0.0

Time: 39.6s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r2532712 = 1.0;
        double r2532713 = x;
        double r2532714 = r2532712 / r2532713;
        double r2532715 = tan(r2532713);
        double r2532716 = r2532712 / r2532715;
        double r2532717 = r2532714 - r2532716;
        return r2532717;
}

↓

double f(double x) {
        double r2532718 = x;
        double r2532719 = 5.0;
        double r2532720 = pow(r2532718, r2532719);
        double r2532721 = 0.0021164021164021165;
        double r2532722 = 0.022222222222222223;
        double r2532723 = r2532722 * r2532718;
        double r2532724 = r2532723 * r2532718;
        double r2532725 = 0.3333333333333333;
        double r2532726 = r2532724 - r2532725;
        double r2532727 = 0.1111111111111111;
        double r2532728 = fma(r2532724, r2532726, r2532727);
        double r2532729 = r2532718 * r2532718;
        double r2532730 = 1.0973936899862826e-05;
        double r2532731 = r2532729 * r2532730;
        double r2532732 = r2532729 * r2532731;
        double r2532733 = 0.037037037037037035;
        double r2532734 = fma(r2532732, r2532729, r2532733);
        double r2532735 = r2532728 / r2532734;
        double r2532736 = r2532718 / r2532735;
        double r2532737 = fma(r2532720, r2532721, r2532736);
        return r2532737;
}

Error

Bits error versus x

Target

Original	59.8
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 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. Simplified 0.3

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

5. Applied flip3-+ 1.2

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

6. Applied associate-*r/ 1.1

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

7. Simplified 0.3

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

9. Applied associate-/l* 0.0

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

10. Simplified 0.0

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

11. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(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))))