Average Error: 58.3 → 0.6
Time: 5.2s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}
double f(double x) {
        double r59247 = x;
        double r59248 = exp(r59247);
        double r59249 = -r59247;
        double r59250 = exp(r59249);
        double r59251 = r59248 - r59250;
        double r59252 = 2.0;
        double r59253 = r59251 / r59252;
        return r59253;
}
↓
double f(double x) {
        double r59254 = 0.3333333333333333;
        double r59255 = x;
        double r59256 = 3.0;
        double r59257 = pow(r59255, r59256);
        double r59258 = r59254 * r59257;
        double r59259 = 0.016666666666666666;
        double r59260 = 5.0;
        double r59261 = pow(r59255, r59260);
        double r59262 = r59259 * r59261;
        double r59263 = r59258 + r59262;
        double r59264 = 2.0;
        double r59265 = r59264 * r59255;
        double r59266 = r59263 + r59265;
        double r59267 = 2.0;
        double r59268 = r59266 / r59267;
        return r59268;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.3

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

  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. Using strategy rm

  4. Applied associate-+r+0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2020033 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))