invcot (example 3.9)

Average Error: 59.8 → 0.0

Time: 33.5s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r1878304 = 1.0;
        double r1878305 = x;
        double r1878306 = r1878304 / r1878305;
        double r1878307 = tan(r1878305);
        double r1878308 = r1878304 / r1878307;
        double r1878309 = r1878306 - r1878308;
        return r1878309;
}

↓

double f(double x) {
        double r1878310 = x;
        double r1878311 = 5.0;
        double r1878312 = pow(r1878310, r1878311);
        double r1878313 = 0.0021164021164021165;
        double r1878314 = 0.1111111111111111;
        double r1878315 = 0.3333333333333333;
        double r1878316 = 0.022222222222222223;
        double r1878317 = r1878316 * r1878310;
        double r1878318 = r1878310 * r1878317;
        double r1878319 = r1878315 - r1878318;
        double r1878320 = r1878319 * r1878318;
        double r1878321 = r1878314 - r1878320;
        double r1878322 = r1878318 * r1878318;
        double r1878323 = 0.037037037037037035;
        double r1878324 = fma(r1878318, r1878322, r1878323);
        double r1878325 = r1878321 / r1878324;
        double r1878326 = r1878310 / r1878325;
        double r1878327 = fma(r1878312, r1878313, r1878326);
        return r1878327;
}

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

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({x}^{5}\right), \frac{2}{945}, \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right) + \frac{1}{3}\right)\right)\right)}\]
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. Simplified 0.0

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

11. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019133 +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))))