Average Error: 58.0 → 0.6
Time: 3.4s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}
double f(double x) {
        double r63310 = x;
        double r63311 = exp(r63310);
        double r63312 = -r63310;
        double r63313 = exp(r63312);
        double r63314 = r63311 - r63313;
        double r63315 = 2.0;
        double r63316 = r63314 / r63315;
        return r63316;
}
↓
double f(double x) {
        double r63317 = 0.3333333333333333;
        double r63318 = x;
        double r63319 = 3.0;
        double r63320 = pow(r63318, r63319);
        double r63321 = r63317 * r63320;
        double r63322 = 0.016666666666666666;
        double r63323 = 5.0;
        double r63324 = pow(r63318, r63323);
        double r63325 = r63322 * r63324;
        double r63326 = 2.0;
        double r63327 = r63326 * r63318;
        double r63328 = r63325 + r63327;
        double r63329 = r63321 + r63328;
        double r63330 = 2.0;
        double r63331 = r63329 / r63330;
        return r63331;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.0

    \[\frac{e^{x} - e^{-x}}{2}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2}\]

  3. Final simplification0.6

    \[\leadsto \frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))