invcot (example 3.9)

Average Error: 59.9 → 0.0

Time: 34.3s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r2401135 = 1.0;
        double r2401136 = x;
        double r2401137 = r2401135 / r2401136;
        double r2401138 = tan(r2401136);
        double r2401139 = r2401135 / r2401138;
        double r2401140 = r2401137 - r2401139;
        return r2401140;
}

↓

double f(double x) {
        double r2401141 = x;
        double r2401142 = 5.0;
        double r2401143 = pow(r2401141, r2401142);
        double r2401144 = 0.0021164021164021165;
        double r2401145 = 0.3333333333333333;
        double r2401146 = 0.022222222222222223;
        double r2401147 = r2401146 * r2401141;
        double r2401148 = r2401141 * r2401147;
        double r2401149 = r2401145 - r2401148;
        double r2401150 = r2401148 * r2401148;
        double r2401151 = fma(r2401149, r2401145, r2401150);
        double r2401152 = 0.037037037037037035;
        double r2401153 = fma(r2401148, r2401150, r2401152);
        double r2401154 = r2401151 / r2401153;
        double r2401155 = r2401141 / r2401154;
        double r2401156 = fma(r2401143, r2401144, r2401155);
        return r2401156;
}

Error

Bits error versus x

Target

Original	59.9
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 59.9

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

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

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

5. Applied flip3-+ 1.3

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

6. Applied associate-*r/ 1.1

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

7. Simplified 0.3

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

9. Applied associate-/l* 0.0

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

10. Simplified 0.0

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

11. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019139 +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))))