Average Error: 58.1 → 0.6
Time: 3.6s
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 r64260 = x;
        double r64261 = exp(r64260);
        double r64262 = -r64260;
        double r64263 = exp(r64262);
        double r64264 = r64261 - r64263;
        double r64265 = 2.0;
        double r64266 = r64264 / r64265;
        return r64266;
}
↓
double f(double x) {
        double r64267 = 0.3333333333333333;
        double r64268 = x;
        double r64269 = 3.0;
        double r64270 = pow(r64268, r64269);
        double r64271 = r64267 * r64270;
        double r64272 = 0.016666666666666666;
        double r64273 = 5.0;
        double r64274 = pow(r64268, r64273);
        double r64275 = r64272 * r64274;
        double r64276 = 2.0;
        double r64277 = r64276 * r64268;
        double r64278 = r64275 + r64277;
        double r64279 = r64271 + r64278;
        double r64280 = 2.0;
        double r64281 = r64279 / r64280;
        return r64281;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.1

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

  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 2019346 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))