Average Error: 58.0 → 0.6
Time: 13.3s
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 r46296 = x;
        double r46297 = exp(r46296);
        double r46298 = -r46296;
        double r46299 = exp(r46298);
        double r46300 = r46297 - r46299;
        double r46301 = 2.0;
        double r46302 = r46300 / r46301;
        return r46302;
}
↓
double f(double x) {
        double r46303 = 0.3333333333333333;
        double r46304 = x;
        double r46305 = 3.0;
        double r46306 = pow(r46304, r46305);
        double r46307 = r46303 * r46306;
        double r46308 = 0.016666666666666666;
        double r46309 = 5.0;
        double r46310 = pow(r46304, r46309);
        double r46311 = r46308 * r46310;
        double r46312 = 2.0;
        double r46313 = r46312 * r46304;
        double r46314 = r46311 + r46313;
        double r46315 = r46307 + r46314;
        double r46316 = 2.0;
        double r46317 = r46315 / r46316;
        return r46317;
}

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 2019325 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))