Average Error: 57.8 → 0.7
Time: 17.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{1}{60} \cdot {x}^{5} + \left(2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{1}{60} \cdot {x}^{5} + \left(2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2}
double f(double x) {
        double r3611332 = x;
        double r3611333 = exp(r3611332);
        double r3611334 = -r3611332;
        double r3611335 = exp(r3611334);
        double r3611336 = r3611333 - r3611335;
        double r3611337 = 2.0;
        double r3611338 = r3611336 / r3611337;
        return r3611338;
}
↓
double f(double x) {
        double r3611339 = 0.016666666666666666;
        double r3611340 = x;
        double r3611341 = 5.0;
        double r3611342 = pow(r3611340, r3611341);
        double r3611343 = r3611339 * r3611342;
        double r3611344 = 2.0;
        double r3611345 = r3611344 * r3611340;
        double r3611346 = r3611340 * r3611340;
        double r3611347 = 0.3333333333333333;
        double r3611348 = r3611346 * r3611347;
        double r3611349 = r3611348 * r3611340;
        double r3611350 = r3611345 + r3611349;
        double r3611351 = r3611343 + r3611350;
        double r3611352 = r3611351 / r3611344;
        return r3611352;
}

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

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

  3. Simplified0.7

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

  5. Applied distribute-lft-in0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))