Average Error: 58.2 → 0.6
Time: 10.2s
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 r46215 = x;
        double r46216 = exp(r46215);
        double r46217 = -r46215;
        double r46218 = exp(r46217);
        double r46219 = r46216 - r46218;
        double r46220 = 2.0;
        double r46221 = r46219 / r46220;
        return r46221;
}
↓
double f(double x) {
        double r46222 = 0.3333333333333333;
        double r46223 = x;
        double r46224 = 3.0;
        double r46225 = pow(r46223, r46224);
        double r46226 = r46222 * r46225;
        double r46227 = 0.016666666666666666;
        double r46228 = 5.0;
        double r46229 = pow(r46223, r46228);
        double r46230 = r46227 * r46229;
        double r46231 = 2.0;
        double r46232 = r46231 * r46223;
        double r46233 = r46230 + r46232;
        double r46234 = r46226 + r46233;
        double r46235 = 2.0;
        double r46236 = r46234 / r46235;
        return r46236;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.2

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

  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 2019208 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))