Average Error: 58.1 → 0.6
Time: 14.5s
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 r1654695 = x;
        double r1654696 = exp(r1654695);
        double r1654697 = -r1654695;
        double r1654698 = exp(r1654697);
        double r1654699 = r1654696 - r1654698;
        double r1654700 = 2.0;
        double r1654701 = r1654699 / r1654700;
        return r1654701;
}
↓
double f(double x) {
        double r1654702 = x;
        double r1654703 = 5.0;
        double r1654704 = pow(r1654702, r1654703);
        double r1654705 = 0.016666666666666666;
        double r1654706 = r1654704 * r1654705;
        double r1654707 = 2.0;
        double r1654708 = 0.3333333333333333;
        double r1654709 = r1654702 * r1654702;
        double r1654710 = r1654708 * r1654709;
        double r1654711 = r1654707 + r1654710;
        double r1654712 = r1654711 * r1654702;
        double r1654713 = r1654706 + r1654712;
        double r1654714 = r1654713 / r1654707;
        return r1654714;
}

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

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

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