Average Error: 59.8 → 0.3
Time: 29.3s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left({x}^{5}, \frac{2}{945}, \mathsf{fma}\left(\frac{1}{45}, x \cdot x, \frac{1}{3}\right) \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left({x}^{5}, \frac{2}{945}, \mathsf{fma}\left(\frac{1}{45}, x \cdot x, \frac{1}{3}\right) \cdot x\right)
double f(double x) {
        double r1835132 = 1.0;
        double r1835133 = x;
        double r1835134 = r1835132 / r1835133;
        double r1835135 = tan(r1835133);
        double r1835136 = r1835132 / r1835135;
        double r1835137 = r1835134 - r1835136;
        return r1835137;
}
↓
double f(double x) {
        double r1835138 = x;
        double r1835139 = 5.0;
        double r1835140 = pow(r1835138, r1835139);
        double r1835141 = 0.0021164021164021165;
        double r1835142 = 0.022222222222222223;
        double r1835143 = r1835138 * r1835138;
        double r1835144 = 0.3333333333333333;
        double r1835145 = fma(r1835142, r1835143, r1835144);
        double r1835146 = r1835145 * r1835138;
        double r1835147 = fma(r1835140, r1835141, r1835146);
        return r1835147;
}

Error

Bits error versus x

Target

Original59.8
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

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. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 +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))))