Average Error: 57.9 → 0.7
Time: 18.7s
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 r40256 = x;
        double r40257 = exp(r40256);
        double r40258 = -r40256;
        double r40259 = exp(r40258);
        double r40260 = r40257 - r40259;
        double r40261 = 2.0;
        double r40262 = r40260 / r40261;
        return r40262;
}
↓
double f(double x) {
        double r40263 = 0.3333333333333333;
        double r40264 = x;
        double r40265 = 3.0;
        double r40266 = pow(r40264, r40265);
        double r40267 = r40263 * r40266;
        double r40268 = 0.016666666666666666;
        double r40269 = 5.0;
        double r40270 = pow(r40264, r40269);
        double r40271 = r40268 * r40270;
        double r40272 = 2.0;
        double r40273 = r40272 * r40264;
        double r40274 = r40271 + r40273;
        double r40275 = r40267 + r40274;
        double r40276 = 2.0;
        double r40277 = r40275 / r40276;
        return r40277;
}

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 0.7

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

  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 2019306 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))