Average Error: 58.0 → 0.7
Time: 8.9s
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 r38738 = x;
        double r38739 = exp(r38738);
        double r38740 = -r38738;
        double r38741 = exp(r38740);
        double r38742 = r38739 - r38741;
        double r38743 = 2.0;
        double r38744 = r38742 / r38743;
        return r38744;
}
↓
double f(double x) {
        double r38745 = 0.3333333333333333;
        double r38746 = x;
        double r38747 = 3.0;
        double r38748 = pow(r38746, r38747);
        double r38749 = r38745 * r38748;
        double r38750 = 0.016666666666666666;
        double r38751 = 5.0;
        double r38752 = pow(r38746, r38751);
        double r38753 = r38750 * r38752;
        double r38754 = 2.0;
        double r38755 = r38754 * r38746;
        double r38756 = r38753 + r38755;
        double r38757 = r38749 + r38756;
        double r38758 = 2.0;
        double r38759 = r38757 / r38758;
        return r38759;
}

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. 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 2020047 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))