Average Error: 58.0 → 0.7
Time: 21.0s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{{x}^{5}}{60} + \left(\frac{{x}^{3}}{3} + \left(x + x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{{x}^{5}}{60} + \left(\frac{{x}^{3}}{3} + \left(x + x\right)\right)}{2}
double f(double x) {
        double r48252 = x;
        double r48253 = exp(r48252);
        double r48254 = -r48252;
        double r48255 = exp(r48254);
        double r48256 = r48253 - r48255;
        double r48257 = 2.0;
        double r48258 = r48256 / r48257;
        return r48258;
}
↓
double f(double x) {
        double r48259 = x;
        double r48260 = 5.0;
        double r48261 = pow(r48259, r48260);
        double r48262 = 60.0;
        double r48263 = r48261 / r48262;
        double r48264 = 3.0;
        double r48265 = pow(r48259, r48264);
        double r48266 = r48265 / r48264;
        double r48267 = r48259 + r48259;
        double r48268 = r48266 + r48267;
        double r48269 = r48263 + r48268;
        double r48270 = 2.0;
        double r48271 = r48269 / r48270;
        return r48271;
}

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 0.7

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

  3. Simplified0.7

    \[\leadsto \frac{\color{blue}{\frac{{x}^{5}}{60} + \left(\left(\frac{{x}^{3}}{3} + x\right) + x\right)}}{2}\]
  4. Using strategy rm

  5. Applied associate-+l+0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))