Average Error: 59.9 → 0.3
Time: 31.8s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left(0.002116402116402116544841005563171165704262, {x}^{5}, \left(0.02222222222222222307030925492199457949027 \cdot \left(x \cdot x\right) + 0.3333333333333333148296162562473909929395\right) \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left(0.002116402116402116544841005563171165704262, {x}^{5}, \left(0.02222222222222222307030925492199457949027 \cdot \left(x \cdot x\right) + 0.3333333333333333148296162562473909929395\right) \cdot x\right)
double f(double x) {
        double r4920242 = 1.0;
        double r4920243 = x;
        double r4920244 = r4920242 / r4920243;
        double r4920245 = tan(r4920243);
        double r4920246 = r4920242 / r4920245;
        double r4920247 = r4920244 - r4920246;
        return r4920247;
}
↓
double f(double x) {
        double r4920248 = 0.0021164021164021165;
        double r4920249 = x;
        double r4920250 = 5.0;
        double r4920251 = pow(r4920249, r4920250);
        double r4920252 = 0.022222222222222223;
        double r4920253 = r4920249 * r4920249;
        double r4920254 = r4920252 * r4920253;
        double r4920255 = 0.3333333333333333;
        double r4920256 = r4920254 + r4920255;
        double r4920257 = r4920256 * r4920249;
        double r4920258 = fma(r4920248, r4920251, r4920257);
        return r4920258;
}

Error

Bits error versus x

Target

Original59.9
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988065102485279567190446:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.9

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{0.3333333333333333148296162562473909929395 \cdot x + \left(0.02222222222222222307030925492199457949027 \cdot {x}^{3} + 0.002116402116402116544841005563171165704262 \cdot {x}^{5}\right)}\]

  3. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.002116402116402116544841005563171165704262, {x}^{5}, x \cdot \left(0.3333333333333333148296162562473909929395 + 0.02222222222222222307030925492199457949027 \cdot \left(x \cdot x\right)\right)\right)}\]

  4. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(0.002116402116402116544841005563171165704262, {x}^{5}, \left(0.02222222222222222307030925492199457949027 \cdot \left(x \cdot x\right) + 0.3333333333333333148296162562473909929395\right) \cdot x\right)\]

Reproduce

herbie shell --seed 2019170 +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 3.0) (+ 1.0 (/ (* x x) 15.0))) (- (/ 1.0 x) (/ 1.0 (tan x))))

  (- (/ 1.0 x) (/ 1.0 (tan x))))