Average Error: 57.9 → 0.7
Time: 5.2s
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 r62722 = x;
        double r62723 = exp(r62722);
        double r62724 = -r62722;
        double r62725 = exp(r62724);
        double r62726 = r62723 - r62725;
        double r62727 = 2.0;
        double r62728 = r62726 / r62727;
        return r62728;
}
↓
double f(double x) {
        double r62729 = 0.3333333333333333;
        double r62730 = x;
        double r62731 = 3.0;
        double r62732 = pow(r62730, r62731);
        double r62733 = r62729 * r62732;
        double r62734 = 0.016666666666666666;
        double r62735 = 5.0;
        double r62736 = pow(r62730, r62735);
        double r62737 = r62734 * r62736;
        double r62738 = r62733 + r62737;
        double r62739 = 2.0;
        double r62740 = r62739 * r62730;
        double r62741 = r62738 + r62740;
        double r62742 = 2.0;
        double r62743 = r62741 / r62742;
        return r62743;
}

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 0.7

    \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2}\]
  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 2020034 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))