invcot (example 3.9)

Average Error: 59.9 → 0.0

Time: 38.0s

Precision: 64

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

Math

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

↓

\[\mathsf{fma}\left({x}^{5}, \frac{2}{945}, \frac{x}{\frac{\mathsf{fma}\left(\frac{1}{2025}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\frac{1}{3}, \frac{1}{3}, \left(\frac{-1}{45} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3}\right)\right)}{\mathsf{fma}\left(x \cdot x, \left(\left(x \cdot x\right) \cdot \frac{1}{91125}\right) \cdot \left(x \cdot x\right), \frac{1}{27}\right)}}\right)\]

C

double f(double x) {
        double r2866785 = 1.0;
        double r2866786 = x;
        double r2866787 = r2866785 / r2866786;
        double r2866788 = tan(r2866786);
        double r2866789 = r2866785 / r2866788;
        double r2866790 = r2866787 - r2866789;
        return r2866790;
}

↓

double f(double x) {
        double r2866791 = x;
        double r2866792 = 5.0;
        double r2866793 = pow(r2866791, r2866792);
        double r2866794 = 0.0021164021164021165;
        double r2866795 = 0.0004938271604938272;
        double r2866796 = r2866791 * r2866791;
        double r2866797 = r2866796 * r2866796;
        double r2866798 = 0.3333333333333333;
        double r2866799 = -0.022222222222222223;
        double r2866800 = r2866799 * r2866796;
        double r2866801 = r2866800 * r2866798;
        double r2866802 = fma(r2866798, r2866798, r2866801);
        double r2866803 = fma(r2866795, r2866797, r2866802);
        double r2866804 = 1.0973936899862826e-05;
        double r2866805 = r2866796 * r2866804;
        double r2866806 = r2866805 * r2866796;
        double r2866807 = 0.037037037037037035;
        double r2866808 = fma(r2866796, r2866806, r2866807);
        double r2866809 = r2866803 / r2866808;
        double r2866810 = r2866791 / r2866809;
        double r2866811 = fma(r2866793, r2866794, r2866810);
        return r2866811;
}

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

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

5. Applied flip3-+ 1.2

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

6. Applied associate-*r/ 1.1

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

7. Simplified 0.3

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

9. Applied associate-/l* 0.0

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

10. Simplified 0.0

    \[\leadsto \mathsf{fma}\left({x}^{5}, \frac{2}{945}, \frac{x}{\color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2025}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\frac{1}{3}, \frac{1}{3}, \left(\frac{-1}{45} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3}\right)\right)}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{91125}\right), \frac{1}{27}\right)}}}\right)\]

11. Final simplification 0.0

    \[\leadsto \mathsf{fma}\left({x}^{5}, \frac{2}{945}, \frac{x}{\frac{\mathsf{fma}\left(\frac{1}{2025}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\frac{1}{3}, \frac{1}{3}, \left(\frac{-1}{45} \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3}\right)\right)}{\mathsf{fma}\left(x \cdot x, \left(\left(x \cdot x\right) \cdot \frac{1}{91125}\right) \cdot \left(x \cdot x\right), \frac{1}{27}\right)}}\right)\]

Reproduce

herbie shell --seed 2019144 +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))))