Average Error: 58.0 → 0.6
Time: 1.1m
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 r36276 = x;
        double r36277 = exp(r36276);
        double r36278 = -r36276;
        double r36279 = exp(r36278);
        double r36280 = r36277 - r36279;
        double r36281 = 2.0;
        double r36282 = r36280 / r36281;
        return r36282;
}
↓
double f(double x) {
        double r36283 = 0.3333333333333333;
        double r36284 = x;
        double r36285 = 3.0;
        double r36286 = pow(r36284, r36285);
        double r36287 = r36283 * r36286;
        double r36288 = 0.016666666666666666;
        double r36289 = 5.0;
        double r36290 = pow(r36284, r36289);
        double r36291 = r36288 * r36290;
        double r36292 = 2.0;
        double r36293 = r36292 * r36284;
        double r36294 = r36291 + r36293;
        double r36295 = r36287 + r36294;
        double r36296 = 2.0;
        double r36297 = r36295 / r36296;
        return r36297;
}

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 2020045 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))