invcot (example 3.9)

Average Error: 59.9 → 0.0

Time: 31.7s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r2609074 = 1.0;
        double r2609075 = x;
        double r2609076 = r2609074 / r2609075;
        double r2609077 = tan(r2609075);
        double r2609078 = r2609074 / r2609077;
        double r2609079 = r2609076 - r2609078;
        return r2609079;
}

↓

double f(double x) {
        double r2609080 = x;
        double r2609081 = 5.0;
        double r2609082 = pow(r2609080, r2609081);
        double r2609083 = 0.0021164021164021165;
        double r2609084 = 0.1111111111111111;
        double r2609085 = 0.022222222222222223;
        double r2609086 = r2609085 * r2609080;
        double r2609087 = r2609086 * r2609080;
        double r2609088 = 0.3333333333333333;
        double r2609089 = r2609087 * r2609088;
        double r2609090 = r2609084 - r2609089;
        double r2609091 = r2609087 * r2609087;
        double r2609092 = r2609090 + r2609091;
        double r2609093 = 0.037037037037037035;
        double r2609094 = fma(r2609091, r2609087, r2609093);
        double r2609095 = r2609092 / r2609094;
        double r2609096 = r2609080 / r2609095;
        double r2609097 = fma(r2609082, r2609083, r2609096);
        return r2609097;
}

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

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

5. Applied flip3-+ 1.2

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

6. Applied associate-*r/ 1.1

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

7. Simplified 0.3

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

9. Applied associate-/l* 0.0

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

10. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019130 +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))))