Average Error: 60.0 → 0.3
Time: 13.2s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\mathsf{fma}\left(0.02222222222222222307030925492199457949027, {x}^{3}, \mathsf{fma}\left(0.3333333333333333148296162562473909929395, x, 0.002116402116402116544841005563171165704262 \cdot {x}^{5}\right)\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\mathsf{fma}\left(0.02222222222222222307030925492199457949027, {x}^{3}, \mathsf{fma}\left(0.3333333333333333148296162562473909929395, x, 0.002116402116402116544841005563171165704262 \cdot {x}^{5}\right)\right)
double f(double x) {
        double r103596 = 1.0;
        double r103597 = x;
        double r103598 = r103596 / r103597;
        double r103599 = tan(r103597);
        double r103600 = r103596 / r103599;
        double r103601 = r103598 - r103600;
        return r103601;
}
↓
double f(double x) {
        double r103602 = 0.022222222222222223;
        double r103603 = x;
        double r103604 = 3.0;
        double r103605 = pow(r103603, r103604);
        double r103606 = 0.3333333333333333;
        double r103607 = 0.0021164021164021165;
        double r103608 = 5.0;
        double r103609 = pow(r103603, r103608);
        double r103610 = r103607 * r103609;
        double r103611 = fma(r103606, r103603, r103610);
        double r103612 = fma(r103602, r103605, r103611);
        return r103612;
}

Error

Bits error versus x

Target

Original60.0
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 60.0

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Taylor expanded around 0 0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(0.02222222222222222307030925492199457949027, {x}^{3}, \mathsf{fma}\left(0.3333333333333333148296162562473909929395, x, 0.002116402116402116544841005563171165704262 \cdot {x}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :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))))