Average Error: 58.0 → 0.7
Time: 14.2s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{x \cdot 2 + \left({x}^{5} \cdot \frac{1}{60} + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{x \cdot 2 + \left({x}^{5} \cdot \frac{1}{60} + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)}{2}
double f(double x) {
        double r908226 = x;
        double r908227 = exp(r908226);
        double r908228 = -r908226;
        double r908229 = exp(r908228);
        double r908230 = r908227 - r908229;
        double r908231 = 2.0;
        double r908232 = r908230 / r908231;
        return r908232;
}
↓
double f(double x) {
        double r908233 = x;
        double r908234 = 2.0;
        double r908235 = r908233 * r908234;
        double r908236 = 5.0;
        double r908237 = pow(r908233, r908236);
        double r908238 = 0.016666666666666666;
        double r908239 = r908237 * r908238;
        double r908240 = 0.3333333333333333;
        double r908241 = r908233 * r908233;
        double r908242 = r908240 * r908241;
        double r908243 = r908242 * r908233;
        double r908244 = r908239 + r908243;
        double r908245 = r908235 + r908244;
        double r908246 = r908245 / r908234;
        return r908246;
}

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

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

  3. Simplified0.7

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

  5. Applied distribute-lft-in0.7

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

  6. Applied associate-+l+0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019155 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))