Average Error: 58.0 → 0.5
Time: 5.2s
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 r36201 = x;
        double r36202 = exp(r36201);
        double r36203 = -r36201;
        double r36204 = exp(r36203);
        double r36205 = r36202 - r36204;
        double r36206 = 2.0;
        double r36207 = r36205 / r36206;
        return r36207;
}
↓
double f(double x) {
        double r36208 = 0.3333333333333333;
        double r36209 = x;
        double r36210 = 3.0;
        double r36211 = pow(r36209, r36210);
        double r36212 = r36208 * r36211;
        double r36213 = 0.016666666666666666;
        double r36214 = 5.0;
        double r36215 = pow(r36209, r36214);
        double r36216 = r36213 * r36215;
        double r36217 = 2.0;
        double r36218 = r36217 * r36209;
        double r36219 = r36216 + r36218;
        double r36220 = r36212 + r36219;
        double r36221 = 2.0;
        double r36222 = r36220 / r36221;
        return r36222;
}

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 0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2019352 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))