Average Error: 57.8 → 0.7
Time: 4.3s
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 r77051 = x;
        double r77052 = exp(r77051);
        double r77053 = -r77051;
        double r77054 = exp(r77053);
        double r77055 = r77052 - r77054;
        double r77056 = 2.0;
        double r77057 = r77055 / r77056;
        return r77057;
}
↓
double f(double x) {
        double r77058 = 0.3333333333333333;
        double r77059 = x;
        double r77060 = 3.0;
        double r77061 = pow(r77059, r77060);
        double r77062 = r77058 * r77061;
        double r77063 = 0.016666666666666666;
        double r77064 = 5.0;
        double r77065 = pow(r77059, r77064);
        double r77066 = r77063 * r77065;
        double r77067 = 2.0;
        double r77068 = r77067 * r77059;
        double r77069 = r77066 + r77068;
        double r77070 = r77062 + r77069;
        double r77071 = 2.0;
        double r77072 = r77070 / r77071;
        return r77072;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.8

    \[\frac{e^{x} - e^{-x}}{2}\]

  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 2020027 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))