Average Error: 58.3 → 0.5
Time: 7.0s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(x + x\right) + \left({x}^{3} \cdot \frac{1}{3} + {x}^{5} \cdot \frac{1}{60}\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\left(x + x\right) + \left({x}^{3} \cdot \frac{1}{3} + {x}^{5} \cdot \frac{1}{60}\right)}{2}
double f(double x) {
        double r40937 = x;
        double r40938 = exp(r40937);
        double r40939 = -r40937;
        double r40940 = exp(r40939);
        double r40941 = r40938 - r40940;
        double r40942 = 2.0;
        double r40943 = r40941 / r40942;
        return r40943;
}
↓
double f(double x) {
        double r40944 = x;
        double r40945 = r40944 + r40944;
        double r40946 = 3.0;
        double r40947 = pow(r40944, r40946);
        double r40948 = 0.3333333333333333;
        double r40949 = r40947 * r40948;
        double r40950 = 5.0;
        double r40951 = pow(r40944, r40950);
        double r40952 = 0.016666666666666666;
        double r40953 = r40951 * r40952;
        double r40954 = r40949 + r40953;
        double r40955 = r40945 + r40954;
        double r40956 = 2.0;
        double r40957 = r40955 / r40956;
        return r40957;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.3

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))