Average Error: 58.0 → 0.7
Time: 18.0s
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 r39581 = x;
        double r39582 = exp(r39581);
        double r39583 = -r39581;
        double r39584 = exp(r39583);
        double r39585 = r39582 - r39584;
        double r39586 = 2.0;
        double r39587 = r39585 / r39586;
        return r39587;
}
↓
double f(double x) {
        double r39588 = 0.3333333333333333;
        double r39589 = x;
        double r39590 = 3.0;
        double r39591 = pow(r39589, r39590);
        double r39592 = r39588 * r39591;
        double r39593 = 0.016666666666666666;
        double r39594 = 5.0;
        double r39595 = pow(r39589, r39594);
        double r39596 = r39593 * r39595;
        double r39597 = 2.0;
        double r39598 = r39597 * r39589;
        double r39599 = r39596 + r39598;
        double r39600 = r39592 + r39599;
        double r39601 = 2.0;
        double r39602 = r39600 / r39601;
        return r39602;
}

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. 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 2020047 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))