Average Error: 58.1 → 0.6
Time: 16.9s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{{x}^{5} \cdot \frac{1}{60} + \frac{\left(\left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right)\right) + 8\right) \cdot x}{\left(\left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) - \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot 2\right) + 4}}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{{x}^{5} \cdot \frac{1}{60} + \frac{\left(\left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right)\right) + 8\right) \cdot x}{\left(\left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) - \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot 2\right) + 4}}{2}
double f(double x) {
        double r3192838 = x;
        double r3192839 = exp(r3192838);
        double r3192840 = -r3192838;
        double r3192841 = exp(r3192840);
        double r3192842 = r3192839 - r3192841;
        double r3192843 = 2.0;
        double r3192844 = r3192842 / r3192843;
        return r3192844;
}
↓
double f(double x) {
        double r3192845 = x;
        double r3192846 = 5.0;
        double r3192847 = pow(r3192845, r3192846);
        double r3192848 = 0.016666666666666666;
        double r3192849 = r3192847 * r3192848;
        double r3192850 = 0.3333333333333333;
        double r3192851 = r3192845 * r3192850;
        double r3192852 = r3192845 * r3192851;
        double r3192853 = r3192852 * r3192852;
        double r3192854 = r3192852 * r3192853;
        double r3192855 = 8.0;
        double r3192856 = r3192854 + r3192855;
        double r3192857 = r3192856 * r3192845;
        double r3192858 = 2.0;
        double r3192859 = r3192852 * r3192858;
        double r3192860 = r3192853 - r3192859;
        double r3192861 = 4.0;
        double r3192862 = r3192860 + r3192861;
        double r3192863 = r3192857 / r3192862;
        double r3192864 = r3192849 + r3192863;
        double r3192865 = 2.0;
        double r3192866 = r3192864 / r3192865;
        return r3192866;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.1

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

  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

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

  5. Applied flip3-+0.6

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

  6. Applied associate-*r/0.6

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

  7. Simplified0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))