Average Error: 58.1 → 0.6
Time: 17.8s
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 r2696023 = x;
        double r2696024 = exp(r2696023);
        double r2696025 = -r2696023;
        double r2696026 = exp(r2696025);
        double r2696027 = r2696024 - r2696026;
        double r2696028 = 2.0;
        double r2696029 = r2696027 / r2696028;
        return r2696029;
}
↓
double f(double x) {
        double r2696030 = x;
        double r2696031 = 5.0;
        double r2696032 = pow(r2696030, r2696031);
        double r2696033 = 0.016666666666666666;
        double r2696034 = r2696032 * r2696033;
        double r2696035 = 0.3333333333333333;
        double r2696036 = r2696030 * r2696035;
        double r2696037 = r2696030 * r2696036;
        double r2696038 = r2696037 * r2696037;
        double r2696039 = r2696037 * r2696038;
        double r2696040 = 8.0;
        double r2696041 = r2696039 + r2696040;
        double r2696042 = r2696041 * r2696030;
        double r2696043 = 2.0;
        double r2696044 = r2696037 * r2696043;
        double r2696045 = r2696038 - r2696044;
        double r2696046 = 4.0;
        double r2696047 = r2696045 + r2696046;
        double r2696048 = r2696042 / r2696047;
        double r2696049 = r2696034 + r2696048;
        double r2696050 = 2.0;
        double r2696051 = r2696049 / r2696050;
        return r2696051;
}

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))