Average Error: 58.0 → 0.7
Time: 5.4s
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 r68491 = x;
        double r68492 = exp(r68491);
        double r68493 = -r68491;
        double r68494 = exp(r68493);
        double r68495 = r68492 - r68494;
        double r68496 = 2.0;
        double r68497 = r68495 / r68496;
        return r68497;
}
↓
double f(double x) {
        double r68498 = 0.3333333333333333;
        double r68499 = x;
        double r68500 = 3.0;
        double r68501 = pow(r68499, r68500);
        double r68502 = r68498 * r68501;
        double r68503 = 0.016666666666666666;
        double r68504 = 5.0;
        double r68505 = pow(r68499, r68504);
        double r68506 = r68503 * r68505;
        double r68507 = r68502 + r68506;
        double r68508 = 2.0;
        double r68509 = r68508 * r68499;
        double r68510 = r68507 + r68509;
        double r68511 = 2.0;
        double r68512 = r68510 / r68511;
        return r68512;
}

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 2019318 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))