Average Error: 58.0 → 0.6
Time: 4.1s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}
double f(double x) {
        double r54722 = x;
        double r54723 = exp(r54722);
        double r54724 = -r54722;
        double r54725 = exp(r54724);
        double r54726 = r54723 - r54725;
        double r54727 = 2.0;
        double r54728 = r54726 / r54727;
        return r54728;
}
↓
double f(double x) {
        double r54729 = 0.3333333333333333;
        double r54730 = x;
        double r54731 = 3.0;
        double r54732 = pow(r54730, r54731);
        double r54733 = r54729 * r54732;
        double r54734 = 0.016666666666666666;
        double r54735 = 5.0;
        double r54736 = pow(r54730, r54735);
        double r54737 = r54734 * r54736;
        double r54738 = r54733 + r54737;
        double r54739 = 2.0;
        double r54740 = r54739 * r54730;
        double r54741 = r54738 + r54740;
        double r54742 = 2.0;
        double r54743 = r54741 / r54742;
        return r54743;
}

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 0.6

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

  4. Applied associate-+r+0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))