Average Error: 58.0 → 0.7
Time: 4.3s
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 r57269 = x;
        double r57270 = exp(r57269);
        double r57271 = -r57269;
        double r57272 = exp(r57271);
        double r57273 = r57270 - r57272;
        double r57274 = 2.0;
        double r57275 = r57273 / r57274;
        return r57275;
}
↓
double f(double x) {
        double r57276 = 0.3333333333333333;
        double r57277 = x;
        double r57278 = 3.0;
        double r57279 = pow(r57277, r57278);
        double r57280 = r57276 * r57279;
        double r57281 = 0.016666666666666666;
        double r57282 = 5.0;
        double r57283 = pow(r57277, r57282);
        double r57284 = r57281 * r57283;
        double r57285 = 2.0;
        double r57286 = r57285 * r57277;
        double r57287 = r57284 + r57286;
        double r57288 = r57280 + r57287;
        double r57289 = 2.0;
        double r57290 = r57288 / r57289;
        return r57290;
}

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 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2019354 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))