Average Error: 59.9 → 0.3
Time: 31.1s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
0.02222222222222222307030925492199457949027 \cdot {x}^{3} + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + 0.3333333333333333148296162562473909929395 \cdot x\right)
double f(double x) {
        double r102538 = 1.0;
        double r102539 = x;
        double r102540 = r102538 / r102539;
        double r102541 = tan(r102539);
        double r102542 = r102538 / r102541;
        double r102543 = r102540 - r102542;
        return r102543;
}
↓
double f(double x) {
        double r102544 = 0.022222222222222223;
        double r102545 = x;
        double r102546 = 3.0;
        double r102547 = pow(r102545, r102546);
        double r102548 = r102544 * r102547;
        double r102549 = 0.0021164021164021165;
        double r102550 = 5.0;
        double r102551 = pow(r102545, r102550);
        double r102552 = r102549 * r102551;
        double r102553 = 0.3333333333333333;
        double r102554 = r102553 * r102545;
        double r102555 = r102552 + r102554;
        double r102556 = r102548 + r102555;
        return r102556;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 
(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))))