Average Error: 58.1 → 0.6
Time: 36.5s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{1}{60} \cdot {x}^{5} + \left(x \cdot \left(x \cdot \frac{1}{3}\right) + 2\right) \cdot x}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{1}{60} \cdot {x}^{5} + \left(x \cdot \left(x \cdot \frac{1}{3}\right) + 2\right) \cdot x}{2}
double f(double x) {
        double r5681890 = x;
        double r5681891 = exp(r5681890);
        double r5681892 = -r5681890;
        double r5681893 = exp(r5681892);
        double r5681894 = r5681891 - r5681893;
        double r5681895 = 2.0;
        double r5681896 = r5681894 / r5681895;
        return r5681896;
}
↓
double f(double x) {
        double r5681897 = 0.016666666666666666;
        double r5681898 = x;
        double r5681899 = 5.0;
        double r5681900 = pow(r5681898, r5681899);
        double r5681901 = r5681897 * r5681900;
        double r5681902 = 0.3333333333333333;
        double r5681903 = r5681898 * r5681902;
        double r5681904 = r5681898 * r5681903;
        double r5681905 = 2.0;
        double r5681906 = r5681904 + r5681905;
        double r5681907 = r5681906 * r5681898;
        double r5681908 = r5681901 + r5681907;
        double r5681909 = r5681908 / r5681905;
        return r5681909;
}

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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019121 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))