Average Error: 58.0 → 0.6
Time: 12.1s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{{x}^{5} \cdot \frac{1}{60} + \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{{x}^{5} \cdot \frac{1}{60} + \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x}{2}
double f(double x) {
        double r2675259 = x;
        double r2675260 = exp(r2675259);
        double r2675261 = -r2675259;
        double r2675262 = exp(r2675261);
        double r2675263 = r2675260 - r2675262;
        double r2675264 = 2.0;
        double r2675265 = r2675263 / r2675264;
        return r2675265;
}
↓
double f(double x) {
        double r2675266 = x;
        double r2675267 = 5.0;
        double r2675268 = pow(r2675266, r2675267);
        double r2675269 = 0.016666666666666666;
        double r2675270 = r2675268 * r2675269;
        double r2675271 = 2.0;
        double r2675272 = 0.3333333333333333;
        double r2675273 = r2675266 * r2675266;
        double r2675274 = r2675272 * r2675273;
        double r2675275 = r2675271 + r2675274;
        double r2675276 = r2675275 * r2675266;
        double r2675277 = r2675270 + r2675276;
        double r2675278 = r2675277 / r2675271;
        return r2675278;
}

Error

Bits error versus x

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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

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))