Average Error: 58.1 → 0.7
Time: 10.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}
double f(double x) {
        double r59941 = x;
        double r59942 = exp(r59941);
        double r59943 = -r59941;
        double r59944 = exp(r59943);
        double r59945 = r59942 - r59944;
        double r59946 = 2.0;
        double r59947 = r59945 / r59946;
        return r59947;
}
↓
double f(double x) {
        double r59948 = 0.3333333333333333;
        double r59949 = x;
        double r59950 = 3.0;
        double r59951 = pow(r59949, r59950);
        double r59952 = r59948 * r59951;
        double r59953 = 0.016666666666666666;
        double r59954 = 5.0;
        double r59955 = pow(r59949, r59954);
        double r59956 = r59953 * r59955;
        double r59957 = r59952 + r59956;
        double r59958 = 2.0;
        double r59959 = r59958 * r59949;
        double r59960 = r59957 + r59959;
        double r59961 = 2.0;
        double r59962 = r59960 / r59961;
        return r59962;
}

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 0.7

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

  4. Applied associate-+r+0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019305 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))