Average Error: 58.0 → 0.6
Time: 12.4s
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 r69105 = x;
        double r69106 = exp(r69105);
        double r69107 = -r69105;
        double r69108 = exp(r69107);
        double r69109 = r69106 - r69108;
        double r69110 = 2.0;
        double r69111 = r69109 / r69110;
        return r69111;
}
↓
double f(double x) {
        double r69112 = x;
        double r69113 = r69112 + r69112;
        double r69114 = 3.0;
        double r69115 = pow(r69112, r69114);
        double r69116 = 0.3333333333333333;
        double r69117 = r69115 * r69116;
        double r69118 = 5.0;
        double r69119 = pow(r69112, r69118);
        double r69120 = 0.016666666666666666;
        double r69121 = r69119 * r69120;
        double r69122 = r69117 + r69121;
        double r69123 = r69113 + r69122;
        double r69124 = 2.0;
        double r69125 = r69123 / r69124;
        return r69125;
}

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 2019195 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))