Average Error: 58.1 → 0.6
Time: 4.9s
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 r69073 = x;
        double r69074 = exp(r69073);
        double r69075 = -r69073;
        double r69076 = exp(r69075);
        double r69077 = r69074 - r69076;
        double r69078 = 2.0;
        double r69079 = r69077 / r69078;
        return r69079;
}
↓
double f(double x) {
        double r69080 = 0.3333333333333333;
        double r69081 = x;
        double r69082 = 3.0;
        double r69083 = pow(r69081, r69082);
        double r69084 = r69080 * r69083;
        double r69085 = 0.016666666666666666;
        double r69086 = 5.0;
        double r69087 = pow(r69081, r69086);
        double r69088 = r69085 * r69087;
        double r69089 = 2.0;
        double r69090 = r69089 * r69081;
        double r69091 = r69088 + r69090;
        double r69092 = r69084 + r69091;
        double r69093 = 2.0;
        double r69094 = r69092 / r69093;
        return r69094;
}

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

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

  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 2019356 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))