Average Error: 58.1 → 0.6
Time: 11.4s
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 r3869264 = x;
        double r3869265 = exp(r3869264);
        double r3869266 = -r3869264;
        double r3869267 = exp(r3869266);
        double r3869268 = r3869265 - r3869267;
        double r3869269 = 2.0;
        double r3869270 = r3869268 / r3869269;
        return r3869270;
}
↓
double f(double x) {
        double r3869271 = 0.016666666666666666;
        double r3869272 = x;
        double r3869273 = 5.0;
        double r3869274 = pow(r3869272, r3869273);
        double r3869275 = r3869271 * r3869274;
        double r3869276 = 0.3333333333333333;
        double r3869277 = r3869272 * r3869276;
        double r3869278 = r3869272 * r3869277;
        double r3869279 = 2.0;
        double r3869280 = r3869278 + r3869279;
        double r3869281 = r3869280 * r3869272;
        double r3869282 = r3869275 + r3869281;
        double r3869283 = r3869282 / r3869279;
        return r3869283;
}

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 0.5

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

  3. Simplified0.6

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

  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 2019104 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))