Average Error: 58.1 → 0.6
Time: 23.3s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{{x}^{5} \cdot \frac{1}{60} + x \cdot \left(2 + \left(x \cdot x\right) \cdot \frac{1}{3}\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{{x}^{5} \cdot \frac{1}{60} + x \cdot \left(2 + \left(x \cdot x\right) \cdot \frac{1}{3}\right)}{2}
double f(double x) {
        double r2541696 = x;
        double r2541697 = exp(r2541696);
        double r2541698 = -r2541696;
        double r2541699 = exp(r2541698);
        double r2541700 = r2541697 - r2541699;
        double r2541701 = 2.0;
        double r2541702 = r2541700 / r2541701;
        return r2541702;
}
↓
double f(double x) {
        double r2541703 = x;
        double r2541704 = 5.0;
        double r2541705 = pow(r2541703, r2541704);
        double r2541706 = 0.016666666666666666;
        double r2541707 = r2541705 * r2541706;
        double r2541708 = 2.0;
        double r2541709 = r2541703 * r2541703;
        double r2541710 = 0.3333333333333333;
        double r2541711 = r2541709 * r2541710;
        double r2541712 = r2541708 + r2541711;
        double r2541713 = r2541703 * r2541712;
        double r2541714 = r2541707 + r2541713;
        double r2541715 = r2541714 / r2541708;
        return r2541715;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.1

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

  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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))