Average Error: 58.0 → 0.6
Time: 13.2s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}
double f(double x) {
        double r40171 = x;
        double r40172 = exp(r40171);
        double r40173 = -r40171;
        double r40174 = exp(r40173);
        double r40175 = r40172 - r40174;
        double r40176 = 2.0;
        double r40177 = r40175 / r40176;
        return r40177;
}
↓
double f(double x) {
        double r40178 = 0.3333333333333333;
        double r40179 = x;
        double r40180 = 3.0;
        double r40181 = pow(r40179, r40180);
        double r40182 = r40178 * r40181;
        double r40183 = 0.016666666666666666;
        double r40184 = 5.0;
        double r40185 = pow(r40179, r40184);
        double r40186 = r40183 * r40185;
        double r40187 = 2.0;
        double r40188 = r40187 * r40179;
        double r40189 = r40186 + r40188;
        double r40190 = r40182 + r40189;
        double r40191 = 2.0;
        double r40192 = r40190 / r40191;
        return r40192;
}

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. Final simplification0.6

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

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))