Average Error: 58.0 → 0.6
Time: 20.4s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{x \cdot \left(4 - \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2 - \frac{1}{3} \cdot \left(x \cdot x\right)} + \frac{1}{60} \cdot {x}^{5}}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{x \cdot \left(4 - \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2 - \frac{1}{3} \cdot \left(x \cdot x\right)} + \frac{1}{60} \cdot {x}^{5}}{2}
double f(double x) {
        double r2081719 = x;
        double r2081720 = exp(r2081719);
        double r2081721 = -r2081719;
        double r2081722 = exp(r2081721);
        double r2081723 = r2081720 - r2081722;
        double r2081724 = 2.0;
        double r2081725 = r2081723 / r2081724;
        return r2081725;
}
↓
double f(double x) {
        double r2081726 = x;
        double r2081727 = 4.0;
        double r2081728 = 0.3333333333333333;
        double r2081729 = r2081726 * r2081726;
        double r2081730 = r2081728 * r2081729;
        double r2081731 = r2081730 * r2081730;
        double r2081732 = r2081727 - r2081731;
        double r2081733 = r2081726 * r2081732;
        double r2081734 = 2.0;
        double r2081735 = r2081734 - r2081730;
        double r2081736 = r2081733 / r2081735;
        double r2081737 = 0.016666666666666666;
        double r2081738 = 5.0;
        double r2081739 = pow(r2081726, r2081738);
        double r2081740 = r2081737 * r2081739;
        double r2081741 = r2081736 + r2081740;
        double r2081742 = r2081741 / r2081734;
        return r2081742;
}

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

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

  3. Simplified0.6

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

  5. Applied flip-+0.6

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

  6. Applied associate-*l/0.6

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

  7. Simplified0.6

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

  8. Final simplification0.6

    \[\leadsto \frac{\frac{x \cdot \left(4 - \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2 - \frac{1}{3} \cdot \left(x \cdot x\right)} + \frac{1}{60} \cdot {x}^{5}}{2}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))