Average Error: 58.0 → 0.6
Time: 13.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(x + x\right) + \left({x}^{3} \cdot \frac{1}{3} + {x}^{5} \cdot \frac{1}{60}\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\left(x + x\right) + \left({x}^{3} \cdot \frac{1}{3} + {x}^{5} \cdot \frac{1}{60}\right)}{2}
double f(double x) {
        double r56421 = x;
        double r56422 = exp(r56421);
        double r56423 = -r56421;
        double r56424 = exp(r56423);
        double r56425 = r56422 - r56424;
        double r56426 = 2.0;
        double r56427 = r56425 / r56426;
        return r56427;
}
↓
double f(double x) {
        double r56428 = x;
        double r56429 = r56428 + r56428;
        double r56430 = 3.0;
        double r56431 = pow(r56428, r56430);
        double r56432 = 0.3333333333333333;
        double r56433 = r56431 * r56432;
        double r56434 = 5.0;
        double r56435 = pow(r56428, r56434);
        double r56436 = 0.016666666666666666;
        double r56437 = r56435 * r56436;
        double r56438 = r56433 + r56437;
        double r56439 = r56429 + r56438;
        double r56440 = 2.0;
        double r56441 = r56439 / r56440;
        return r56441;
}

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 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))