Average Error: 58.0 → 0.6
Time: 18.2s
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 r42589 = x;
        double r42590 = exp(r42589);
        double r42591 = -r42589;
        double r42592 = exp(r42591);
        double r42593 = r42590 - r42592;
        double r42594 = 2.0;
        double r42595 = r42593 / r42594;
        return r42595;
}
↓
double f(double x) {
        double r42596 = 0.3333333333333333;
        double r42597 = x;
        double r42598 = 3.0;
        double r42599 = pow(r42597, r42598);
        double r42600 = r42596 * r42599;
        double r42601 = 0.016666666666666666;
        double r42602 = 5.0;
        double r42603 = pow(r42597, r42602);
        double r42604 = r42601 * r42603;
        double r42605 = r42600 + r42604;
        double r42606 = 2.0;
        double r42607 = r42606 * r42597;
        double r42608 = r42605 + r42607;
        double r42609 = 2.0;
        double r42610 = r42608 / r42609;
        return r42610;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.0

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

  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 2019304 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))