Average Error: 58.0 → 0.7
Time: 4.3s
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 r55592 = x;
        double r55593 = exp(r55592);
        double r55594 = -r55592;
        double r55595 = exp(r55594);
        double r55596 = r55593 - r55595;
        double r55597 = 2.0;
        double r55598 = r55596 / r55597;
        return r55598;
}
↓
double f(double x) {
        double r55599 = 0.3333333333333333;
        double r55600 = x;
        double r55601 = 3.0;
        double r55602 = pow(r55600, r55601);
        double r55603 = r55599 * r55602;
        double r55604 = 0.016666666666666666;
        double r55605 = 5.0;
        double r55606 = pow(r55600, r55605);
        double r55607 = r55604 * r55606;
        double r55608 = r55603 + r55607;
        double r55609 = 2.0;
        double r55610 = r55609 * r55600;
        double r55611 = r55608 + r55610;
        double r55612 = 2.0;
        double r55613 = r55611 / r55612;
        return r55613;
}

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 0.7

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

  4. Applied associate-+r+0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))