Average Error: 58.1 → 0.7
Time: 11.0s
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 r57950 = x;
        double r57951 = exp(r57950);
        double r57952 = -r57950;
        double r57953 = exp(r57952);
        double r57954 = r57951 - r57953;
        double r57955 = 2.0;
        double r57956 = r57954 / r57955;
        return r57956;
}
↓
double f(double x) {
        double r57957 = 0.3333333333333333;
        double r57958 = x;
        double r57959 = 3.0;
        double r57960 = pow(r57958, r57959);
        double r57961 = r57957 * r57960;
        double r57962 = 0.016666666666666666;
        double r57963 = 5.0;
        double r57964 = pow(r57958, r57963);
        double r57965 = r57962 * r57964;
        double r57966 = 2.0;
        double r57967 = r57966 * r57958;
        double r57968 = r57965 + r57967;
        double r57969 = r57961 + r57968;
        double r57970 = 2.0;
        double r57971 = r57969 / r57970;
        return r57971;
}

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. Final simplification0.7

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

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))