Average Error: 57.9 → 0.7
Time: 19.5s
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 r2185132 = x;
        double r2185133 = exp(r2185132);
        double r2185134 = -r2185132;
        double r2185135 = exp(r2185134);
        double r2185136 = r2185133 - r2185135;
        double r2185137 = 2.0;
        double r2185138 = r2185136 / r2185137;
        return r2185138;
}
↓
double f(double x) {
        double r2185139 = x;
        double r2185140 = 5.0;
        double r2185141 = pow(r2185139, r2185140);
        double r2185142 = 0.016666666666666666;
        double r2185143 = r2185141 * r2185142;
        double r2185144 = 2.0;
        double r2185145 = r2185139 * r2185139;
        double r2185146 = 0.3333333333333333;
        double r2185147 = r2185145 * r2185146;
        double r2185148 = r2185144 + r2185147;
        double r2185149 = r2185139 * r2185148;
        double r2185150 = r2185143 + r2185149;
        double r2185151 = r2185150 / r2185144;
        return r2185151;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019129 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))