Average Error: 58.0 → 0.6
Time: 4.2s
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 r51302 = x;
        double r51303 = exp(r51302);
        double r51304 = -r51302;
        double r51305 = exp(r51304);
        double r51306 = r51303 - r51305;
        double r51307 = 2.0;
        double r51308 = r51306 / r51307;
        return r51308;
}
↓
double f(double x) {
        double r51309 = 0.3333333333333333;
        double r51310 = x;
        double r51311 = 3.0;
        double r51312 = pow(r51310, r51311);
        double r51313 = r51309 * r51312;
        double r51314 = 0.016666666666666666;
        double r51315 = 5.0;
        double r51316 = pow(r51310, r51315);
        double r51317 = r51314 * r51316;
        double r51318 = 2.0;
        double r51319 = r51318 * r51310;
        double r51320 = r51317 + r51319;
        double r51321 = r51313 + r51320;
        double r51322 = 2.0;
        double r51323 = r51321 / r51322;
        return r51323;
}

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

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

  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 2019347 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))