Average Error: 58.0 → 0.6
Time: 28.9s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{1}{60} \cdot {x}^{5} + \frac{\left(8 + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right)\right)\right) \cdot x}{4 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) - 2 \cdot \log \left(e^{\left(x \cdot x\right) \cdot \frac{1}{3}}\right)\right)}}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{1}{60} \cdot {x}^{5} + \frac{\left(8 + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right)\right)\right) \cdot x}{4 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) - 2 \cdot \log \left(e^{\left(x \cdot x\right) \cdot \frac{1}{3}}\right)\right)}}{2}
double f(double x) {
        double r1645772 = x;
        double r1645773 = exp(r1645772);
        double r1645774 = -r1645772;
        double r1645775 = exp(r1645774);
        double r1645776 = r1645773 - r1645775;
        double r1645777 = 2.0;
        double r1645778 = r1645776 / r1645777;
        return r1645778;
}
↓
double f(double x) {
        double r1645779 = 0.016666666666666666;
        double r1645780 = x;
        double r1645781 = 5.0;
        double r1645782 = pow(r1645780, r1645781);
        double r1645783 = r1645779 * r1645782;
        double r1645784 = 8.0;
        double r1645785 = r1645780 * r1645780;
        double r1645786 = 0.3333333333333333;
        double r1645787 = r1645785 * r1645786;
        double r1645788 = r1645787 * r1645787;
        double r1645789 = r1645787 * r1645788;
        double r1645790 = r1645784 + r1645789;
        double r1645791 = r1645790 * r1645780;
        double r1645792 = 4.0;
        double r1645793 = 2.0;
        double r1645794 = exp(r1645787);
        double r1645795 = log(r1645794);
        double r1645796 = r1645793 * r1645795;
        double r1645797 = r1645788 - r1645796;
        double r1645798 = r1645792 + r1645797;
        double r1645799 = r1645791 / r1645798;
        double r1645800 = r1645783 + r1645799;
        double r1645801 = r1645800 / r1645793;
        return r1645801;
}

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. Simplified0.7

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

  5. Applied flip3-+0.7

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

  6. Applied associate-*l/0.6

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

  7. Simplified0.6

    \[\leadsto \frac{\frac{\color{blue}{\left(8 + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)\right) \cdot x}}{2 \cdot 2 + \left(\left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) - 2 \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)} + \frac{1}{60} \cdot {x}^{5}}{2}\]
  8. Using strategy rm

  9. Applied add-log-exp0.6

    \[\leadsto \frac{\frac{\left(8 + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)\right) \cdot x}{2 \cdot 2 + \left(\left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) - 2 \cdot \color{blue}{\log \left(e^{\frac{1}{3} \cdot \left(x \cdot x\right)}\right)}\right)} + \frac{1}{60} \cdot {x}^{5}}{2}\]

  10. Final simplification0.6

    \[\leadsto \frac{\frac{1}{60} \cdot {x}^{5} + \frac{\left(8 + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right)\right)\right) \cdot x}{4 + \left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) - 2 \cdot \log \left(e^{\left(x \cdot x\right) \cdot \frac{1}{3}}\right)\right)}}{2}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))