Average Error: 58.0 → 0.7
Time: 12.9s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{1}{60} \cdot {x}^{5} + \left(2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{1}{60} \cdot {x}^{5} + \left(2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2}
double f(double x) {
        double r2042334 = x;
        double r2042335 = exp(r2042334);
        double r2042336 = -r2042334;
        double r2042337 = exp(r2042336);
        double r2042338 = r2042335 - r2042337;
        double r2042339 = 2.0;
        double r2042340 = r2042338 / r2042339;
        return r2042340;
}
↓
double f(double x) {
        double r2042341 = 0.016666666666666666;
        double r2042342 = x;
        double r2042343 = 5.0;
        double r2042344 = pow(r2042342, r2042343);
        double r2042345 = r2042341 * r2042344;
        double r2042346 = 2.0;
        double r2042347 = r2042346 * r2042342;
        double r2042348 = r2042342 * r2042342;
        double r2042349 = 0.3333333333333333;
        double r2042350 = r2042348 * r2042349;
        double r2042351 = r2042350 * r2042342;
        double r2042352 = r2042347 + r2042351;
        double r2042353 = r2042345 + r2042352;
        double r2042354 = r2042353 / r2042346;
        return r2042354;
}

Error

Bits error versus x

Try it out

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 0.7

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

  3. Simplified0.7

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

  5. Applied distribute-lft-in0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))