Average Error: 59.9 → 0.0
Time: 33.0s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left({x}^{5}, \frac{2}{945}, \frac{x}{\frac{1}{\mathsf{fma}\left(\frac{1}{45} \cdot x, x, \frac{1}{3}\right)}}\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left({x}^{5}, \frac{2}{945}, \frac{x}{\frac{1}{\mathsf{fma}\left(\frac{1}{45} \cdot x, x, \frac{1}{3}\right)}}\right)
double f(double x) {
        double r2593887 = 1.0;
        double r2593888 = x;
        double r2593889 = r2593887 / r2593888;
        double r2593890 = tan(r2593888);
        double r2593891 = r2593887 / r2593890;
        double r2593892 = r2593889 - r2593891;
        return r2593892;
}
↓
double f(double x) {
        double r2593893 = x;
        double r2593894 = 5.0;
        double r2593895 = pow(r2593893, r2593894);
        double r2593896 = 0.0021164021164021165;
        double r2593897 = 1.0;
        double r2593898 = 0.022222222222222223;
        double r2593899 = r2593898 * r2593893;
        double r2593900 = 0.3333333333333333;
        double r2593901 = fma(r2593899, r2593893, r2593900);
        double r2593902 = r2593897 / r2593901;
        double r2593903 = r2593893 / r2593902;
        double r2593904 = fma(r2593895, r2593896, r2593903);
        return r2593904;
}

Error

Bits error versus x

Target

Original59.9
Target0.1
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

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. Simplified0.3

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

  5. Applied flip-+0.3

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

  6. Applied associate-*r/0.3

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

  8. Applied associate-/l*0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019146 +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))))