Average Error: 58.3 → 0.5
Time: 37.5s
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 r2289429 = x;
        double r2289430 = exp(r2289429);
        double r2289431 = -r2289429;
        double r2289432 = exp(r2289431);
        double r2289433 = r2289430 - r2289432;
        double r2289434 = 2.0;
        double r2289435 = r2289433 / r2289434;
        return r2289435;
}
↓
double f(double x) {
        double r2289436 = x;
        double r2289437 = 5.0;
        double r2289438 = pow(r2289436, r2289437);
        double r2289439 = 0.016666666666666666;
        double r2289440 = r2289438 * r2289439;
        double r2289441 = 2.0;
        double r2289442 = r2289436 * r2289436;
        double r2289443 = 0.3333333333333333;
        double r2289444 = r2289442 * r2289443;
        double r2289445 = r2289441 + r2289444;
        double r2289446 = r2289436 * r2289445;
        double r2289447 = r2289440 + r2289446;
        double r2289448 = r2289447 / r2289441;
        return r2289448;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.3

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019149 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))