Average Error: 58.1 → 0.7
Time: 4.5s
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 r62070 = x;
        double r62071 = exp(r62070);
        double r62072 = -r62070;
        double r62073 = exp(r62072);
        double r62074 = r62071 - r62073;
        double r62075 = 2.0;
        double r62076 = r62074 / r62075;
        return r62076;
}
↓
double f(double x) {
        double r62077 = 0.3333333333333333;
        double r62078 = x;
        double r62079 = 3.0;
        double r62080 = pow(r62078, r62079);
        double r62081 = r62077 * r62080;
        double r62082 = 0.016666666666666666;
        double r62083 = 5.0;
        double r62084 = pow(r62078, r62083);
        double r62085 = r62082 * r62084;
        double r62086 = 2.0;
        double r62087 = r62086 * r62078;
        double r62088 = r62085 + r62087;
        double r62089 = r62081 + r62088;
        double r62090 = 2.0;
        double r62091 = r62089 / r62090;
        return r62091;
}

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 2020059 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))