Average Error: 57.9 → 0.7
Time: 4.1s
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 r57337 = x;
        double r57338 = exp(r57337);
        double r57339 = -r57337;
        double r57340 = exp(r57339);
        double r57341 = r57338 - r57340;
        double r57342 = 2.0;
        double r57343 = r57341 / r57342;
        return r57343;
}
↓
double f(double x) {
        double r57344 = 0.3333333333333333;
        double r57345 = x;
        double r57346 = 3.0;
        double r57347 = pow(r57345, r57346);
        double r57348 = r57344 * r57347;
        double r57349 = 0.016666666666666666;
        double r57350 = 5.0;
        double r57351 = pow(r57345, r57350);
        double r57352 = r57349 * r57351;
        double r57353 = 2.0;
        double r57354 = r57353 * r57345;
        double r57355 = r57352 + r57354;
        double r57356 = r57348 + r57355;
        double r57357 = 2.0;
        double r57358 = r57356 / r57357;
        return r57358;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

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

  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 2020062 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))