Average Error: 57.9 → 0.6
Time: 10.1s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{{x}^{5} \cdot \frac{1}{60} + \left(2 \cdot x + \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{{x}^{5} \cdot \frac{1}{60} + \left(2 \cdot x + \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot x\right)}{2}
double f(double x) {
        double r2607241 = x;
        double r2607242 = exp(r2607241);
        double r2607243 = -r2607241;
        double r2607244 = exp(r2607243);
        double r2607245 = r2607242 - r2607244;
        double r2607246 = 2.0;
        double r2607247 = r2607245 / r2607246;
        return r2607247;
}
↓
double f(double x) {
        double r2607248 = x;
        double r2607249 = 5.0;
        double r2607250 = pow(r2607248, r2607249);
        double r2607251 = 0.016666666666666666;
        double r2607252 = r2607250 * r2607251;
        double r2607253 = 2.0;
        double r2607254 = r2607253 * r2607248;
        double r2607255 = 0.3333333333333333;
        double r2607256 = r2607248 * r2607255;
        double r2607257 = r2607248 * r2607256;
        double r2607258 = r2607257 * r2607248;
        double r2607259 = r2607254 + r2607258;
        double r2607260 = r2607252 + r2607259;
        double r2607261 = 2.0;
        double r2607262 = r2607260 / r2607261;
        return r2607262;
}

Error

Bits error versus x

Try it out

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 0.6

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

  3. Simplified0.6

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

  4. Taylor expanded around 0 0.6

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

  5. Simplified0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))