invcot (example 3.9)

Average Error: 59.8 → 0.0

Time: 32.4s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r2679341 = 1.0;
        double r2679342 = x;
        double r2679343 = r2679341 / r2679342;
        double r2679344 = tan(r2679342);
        double r2679345 = r2679341 / r2679344;
        double r2679346 = r2679343 - r2679345;
        return r2679346;
}

↓

double f(double x) {
        double r2679347 = 0.0021164021164021165;
        double r2679348 = x;
        double r2679349 = 5.0;
        double r2679350 = pow(r2679348, r2679349);
        double r2679351 = -0.3333333333333333;
        double r2679352 = 0.022222222222222223;
        double r2679353 = r2679352 * r2679348;
        double r2679354 = r2679348 * r2679353;
        double r2679355 = r2679354 * r2679354;
        double r2679356 = fma(r2679351, r2679354, r2679355);
        double r2679357 = 0.1111111111111111;
        double r2679358 = r2679356 + r2679357;
        double r2679359 = 0.037037037037037035;
        double r2679360 = fma(r2679355, r2679354, r2679359);
        double r2679361 = r2679358 / r2679360;
        double r2679362 = r2679348 / r2679361;
        double r2679363 = fma(r2679347, r2679350, r2679362);
        return r2679363;
}

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

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

5. Applied flip3-+ 1.2

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

6. Applied associate-*r/ 1.1

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

7. Simplified 0.3

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

9. Applied associate-/l* 0.0

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

10. Simplified 0.0

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

11. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019129 +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))))