Average Error: 57.9 → 0.6
Time: 8.5s
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 r39116 = x;
        double r39117 = exp(r39116);
        double r39118 = -r39116;
        double r39119 = exp(r39118);
        double r39120 = r39117 - r39119;
        double r39121 = 2.0;
        double r39122 = r39120 / r39121;
        return r39122;
}
↓
double f(double x) {
        double r39123 = 0.3333333333333333;
        double r39124 = x;
        double r39125 = 3.0;
        double r39126 = pow(r39124, r39125);
        double r39127 = r39123 * r39126;
        double r39128 = 0.016666666666666666;
        double r39129 = 5.0;
        double r39130 = pow(r39124, r39129);
        double r39131 = r39128 * r39130;
        double r39132 = r39127 + r39131;
        double r39133 = 2.0;
        double r39134 = r39133 * r39124;
        double r39135 = r39132 + r39134;
        double r39136 = 2.0;
        double r39137 = r39135 / r39136;
        return r39137;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

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

  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. Using strategy rm

  4. Applied associate-+r+0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019209 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))