Average Error: 59.9 → 0.3
Time: 28.6s
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 r110928 = 1.0;
        double r110929 = x;
        double r110930 = r110928 / r110929;
        double r110931 = tan(r110929);
        double r110932 = r110928 / r110931;
        double r110933 = r110930 - r110932;
        return r110933;
}
↓
double f(double x) {
        double r110934 = 0.022222222222222223;
        double r110935 = x;
        double r110936 = 3.0;
        double r110937 = pow(r110935, r110936);
        double r110938 = 0.3333333333333333;
        double r110939 = 0.0021164021164021165;
        double r110940 = 5.0;
        double r110941 = pow(r110935, r110940);
        double r110942 = r110939 * r110941;
        double r110943 = fma(r110938, r110935, r110942);
        double r110944 = fma(r110934, r110937, r110943);
        return r110944;
}

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

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

  4. Taylor expanded around 0 0.3

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

  5. Simplified0.3

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

  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 2019194 +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))))