Average Error: 57.9 → 0.7
Time: 17.9s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{x \cdot \left(2 + \left(x \cdot x\right) \cdot \frac{1}{3}\right) + \frac{1}{60} \cdot {x}^{5}}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{x \cdot \left(2 + \left(x \cdot x\right) \cdot \frac{1}{3}\right) + \frac{1}{60} \cdot {x}^{5}}{2}
double f(double x) {
        double r4734890 = x;
        double r4734891 = exp(r4734890);
        double r4734892 = -r4734890;
        double r4734893 = exp(r4734892);
        double r4734894 = r4734891 - r4734893;
        double r4734895 = 2.0;
        double r4734896 = r4734894 / r4734895;
        return r4734896;
}
↓
double f(double x) {
        double r4734897 = x;
        double r4734898 = 2.0;
        double r4734899 = r4734897 * r4734897;
        double r4734900 = 0.3333333333333333;
        double r4734901 = r4734899 * r4734900;
        double r4734902 = r4734898 + r4734901;
        double r4734903 = r4734897 * r4734902;
        double r4734904 = 0.016666666666666666;
        double r4734905 = 5.0;
        double r4734906 = pow(r4734897, r4734905);
        double r4734907 = r4734904 * r4734906;
        double r4734908 = r4734903 + r4734907;
        double r4734909 = 2.0;
        double r4734910 = r4734908 / r4734909;
        return r4734910;
}

Error

Bits error versus x

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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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019173 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))