Average Error: 57.9 → 0.6
Time: 34.0s
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 r67220 = x;
        double r67221 = exp(r67220);
        double r67222 = -r67220;
        double r67223 = exp(r67222);
        double r67224 = r67221 - r67223;
        double r67225 = 2.0;
        double r67226 = r67224 / r67225;
        return r67226;
}
↓
double f(double x) {
        double r67227 = 0.3333333333333333;
        double r67228 = x;
        double r67229 = 3.0;
        double r67230 = pow(r67228, r67229);
        double r67231 = r67227 * r67230;
        double r67232 = 0.016666666666666666;
        double r67233 = 5.0;
        double r67234 = pow(r67228, r67233);
        double r67235 = r67232 * r67234;
        double r67236 = 2.0;
        double r67237 = r67236 * r67228;
        double r67238 = r67235 + r67237;
        double r67239 = r67231 + r67238;
        double r67240 = 2.0;
        double r67241 = r67239 / r67240;
        return r67241;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

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

  2. Taylor expanded around 0 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2019294 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))