Average Error: 58.0 → 0.6
Time: 5.8s
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 r68481 = x;
        double r68482 = exp(r68481);
        double r68483 = -r68481;
        double r68484 = exp(r68483);
        double r68485 = r68482 - r68484;
        double r68486 = 2.0;
        double r68487 = r68485 / r68486;
        return r68487;
}
↓
double f(double x) {
        double r68488 = 0.3333333333333333;
        double r68489 = x;
        double r68490 = 3.0;
        double r68491 = pow(r68489, r68490);
        double r68492 = r68488 * r68491;
        double r68493 = 0.016666666666666666;
        double r68494 = 5.0;
        double r68495 = pow(r68489, r68494);
        double r68496 = r68493 * r68495;
        double r68497 = 2.0;
        double r68498 = r68497 * r68489;
        double r68499 = r68496 + r68498;
        double r68500 = r68492 + r68499;
        double r68501 = 2.0;
        double r68502 = r68500 / r68501;
        return r68502;
}

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. Final simplification0.6

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))