Average Error: 58.0 → 0.7
Time: 55.3s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}
double f(double x) {
        double r40420 = x;
        double r40421 = exp(r40420);
        double r40422 = -r40420;
        double r40423 = exp(r40422);
        double r40424 = r40421 - r40423;
        double r40425 = 2.0;
        double r40426 = r40424 / r40425;
        return r40426;
}
↓
double f(double x) {
        double r40427 = 0.3333333333333333;
        double r40428 = x;
        double r40429 = 3.0;
        double r40430 = pow(r40428, r40429);
        double r40431 = r40427 * r40430;
        double r40432 = 0.016666666666666666;
        double r40433 = 5.0;
        double r40434 = pow(r40428, r40433);
        double r40435 = r40432 * r40434;
        double r40436 = 2.0;
        double r40437 = r40436 * r40428;
        double r40438 = r40435 + r40437;
        double r40439 = r40431 + r40438;
        double r40440 = 2.0;
        double r40441 = r40439 / r40440;
        return r40441;
}

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

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))