Average Error: 57.8 → 0.7
Time: 4.9s
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 r33356 = x;
        double r33357 = exp(r33356);
        double r33358 = -r33356;
        double r33359 = exp(r33358);
        double r33360 = r33357 - r33359;
        double r33361 = 2.0;
        double r33362 = r33360 / r33361;
        return r33362;
}
↓
double f(double x) {
        double r33363 = 0.3333333333333333;
        double r33364 = x;
        double r33365 = 3.0;
        double r33366 = pow(r33364, r33365);
        double r33367 = r33363 * r33366;
        double r33368 = 0.016666666666666666;
        double r33369 = 5.0;
        double r33370 = pow(r33364, r33369);
        double r33371 = r33368 * r33370;
        double r33372 = 2.0;
        double r33373 = r33372 * r33364;
        double r33374 = r33371 + r33373;
        double r33375 = r33367 + r33374;
        double r33376 = 2.0;
        double r33377 = r33375 / r33376;
        return r33377;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.8

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

  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. Final simplification0.7

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

Reproduce

herbie shell --seed 2020100 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))