Average Error: 58.0 → 0.7
Time: 4.1s
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 r65445 = x;
        double r65446 = exp(r65445);
        double r65447 = -r65445;
        double r65448 = exp(r65447);
        double r65449 = r65446 - r65448;
        double r65450 = 2.0;
        double r65451 = r65449 / r65450;
        return r65451;
}
↓
double f(double x) {
        double r65452 = 0.3333333333333333;
        double r65453 = x;
        double r65454 = 3.0;
        double r65455 = pow(r65453, r65454);
        double r65456 = r65452 * r65455;
        double r65457 = 0.016666666666666666;
        double r65458 = 5.0;
        double r65459 = pow(r65453, r65458);
        double r65460 = r65457 * r65459;
        double r65461 = 2.0;
        double r65462 = r65461 * r65453;
        double r65463 = r65460 + r65462;
        double r65464 = r65456 + r65463;
        double r65465 = 2.0;
        double r65466 = r65464 / r65465;
        return r65466;
}

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 2020036 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))