Average Error: 58.0 → 0.7
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 r2349125 = x;
        double r2349126 = exp(r2349125);
        double r2349127 = -r2349125;
        double r2349128 = exp(r2349127);
        double r2349129 = r2349126 - r2349128;
        double r2349130 = 2.0;
        double r2349131 = r2349129 / r2349130;
        return r2349131;
}
↓
double f(double x) {
        double r2349132 = x;
        double r2349133 = 5.0;
        double r2349134 = pow(r2349132, r2349133);
        double r2349135 = 0.016666666666666666;
        double r2349136 = r2349134 * r2349135;
        double r2349137 = 2.0;
        double r2349138 = r2349132 * r2349132;
        double r2349139 = 0.3333333333333333;
        double r2349140 = r2349138 * r2349139;
        double r2349141 = r2349137 + r2349140;
        double r2349142 = r2349132 * r2349141;
        double r2349143 = r2349136 + r2349142;
        double r2349144 = r2349143 / r2349137;
        return r2349144;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.0

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

  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. Simplified0.7

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

  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 2019139 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))