Average Error: 58.1 → 0.7
Time: 33.0s
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 r45104 = x;
        double r45105 = exp(r45104);
        double r45106 = -r45104;
        double r45107 = exp(r45106);
        double r45108 = r45105 - r45107;
        double r45109 = 2.0;
        double r45110 = r45108 / r45109;
        return r45110;
}
↓
double f(double x) {
        double r45111 = 0.3333333333333333;
        double r45112 = x;
        double r45113 = 3.0;
        double r45114 = pow(r45112, r45113);
        double r45115 = r45111 * r45114;
        double r45116 = 0.016666666666666666;
        double r45117 = 5.0;
        double r45118 = pow(r45112, r45117);
        double r45119 = r45116 * r45118;
        double r45120 = 2.0;
        double r45121 = r45120 * r45112;
        double r45122 = r45119 + r45121;
        double r45123 = r45115 + r45122;
        double r45124 = 2.0;
        double r45125 = r45123 / r45124;
        return r45125;
}

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 2019298 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))