Average Error: 57.9 → 0.6
Time: 11.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(\frac{1}{60} \cdot {x}^{5} + \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + 2 \cdot x}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\left(\frac{1}{60} \cdot {x}^{5} + \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + 2 \cdot x}{2}
double f(double x) {
        double r3523097 = x;
        double r3523098 = exp(r3523097);
        double r3523099 = -r3523097;
        double r3523100 = exp(r3523099);
        double r3523101 = r3523098 - r3523100;
        double r3523102 = 2.0;
        double r3523103 = r3523101 / r3523102;
        return r3523103;
}
↓
double f(double x) {
        double r3523104 = 0.016666666666666666;
        double r3523105 = x;
        double r3523106 = 5.0;
        double r3523107 = pow(r3523105, r3523106);
        double r3523108 = r3523104 * r3523107;
        double r3523109 = 0.3333333333333333;
        double r3523110 = r3523105 * r3523105;
        double r3523111 = r3523110 * r3523105;
        double r3523112 = r3523109 * r3523111;
        double r3523113 = r3523108 + r3523112;
        double r3523114 = 2.0;
        double r3523115 = r3523114 * r3523105;
        double r3523116 = r3523113 + r3523115;
        double r3523117 = 2.0;
        double r3523118 = r3523116 / r3523117;
        return r3523118;
}

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

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

  3. Simplified0.6

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

  4. Taylor expanded around 0 0.6

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

  5. Simplified0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))