Average Error: 58.1 → 0.6
Time: 15.6s
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 r54490 = x;
        double r54491 = exp(r54490);
        double r54492 = -r54490;
        double r54493 = exp(r54492);
        double r54494 = r54491 - r54493;
        double r54495 = 2.0;
        double r54496 = r54494 / r54495;
        return r54496;
}
↓
double f(double x) {
        double r54497 = 0.3333333333333333;
        double r54498 = x;
        double r54499 = 3.0;
        double r54500 = pow(r54498, r54499);
        double r54501 = r54497 * r54500;
        double r54502 = 0.016666666666666666;
        double r54503 = 5.0;
        double r54504 = pow(r54498, r54503);
        double r54505 = r54502 * r54504;
        double r54506 = r54501 + r54505;
        double r54507 = 2.0;
        double r54508 = r54507 * r54498;
        double r54509 = r54506 + r54508;
        double r54510 = 2.0;
        double r54511 = r54509 / r54510;
        return r54511;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.1

    \[\frac{e^{x} - e^{-x}}{2}\]

  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. Using strategy rm

  4. Applied associate-+r+0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))