Average Error: 58.1 → 0.6
Time: 8.6s
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 r55497 = x;
        double r55498 = exp(r55497);
        double r55499 = -r55497;
        double r55500 = exp(r55499);
        double r55501 = r55498 - r55500;
        double r55502 = 2.0;
        double r55503 = r55501 / r55502;
        return r55503;
}
↓
double f(double x) {
        double r55504 = 0.3333333333333333;
        double r55505 = x;
        double r55506 = 3.0;
        double r55507 = pow(r55505, r55506);
        double r55508 = r55504 * r55507;
        double r55509 = 0.016666666666666666;
        double r55510 = 5.0;
        double r55511 = pow(r55505, r55510);
        double r55512 = r55509 * r55511;
        double r55513 = r55508 + r55512;
        double r55514 = 2.0;
        double r55515 = r55514 * r55505;
        double r55516 = r55513 + r55515;
        double r55517 = 2.0;
        double r55518 = r55516 / r55517;
        return r55518;
}

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 2019350 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))