Average Error: 58.0 → 0.7
Time: 11.9s
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 r44356 = x;
        double r44357 = exp(r44356);
        double r44358 = -r44356;
        double r44359 = exp(r44358);
        double r44360 = r44357 - r44359;
        double r44361 = 2.0;
        double r44362 = r44360 / r44361;
        return r44362;
}
↓
double f(double x) {
        double r44363 = 0.3333333333333333;
        double r44364 = x;
        double r44365 = 3.0;
        double r44366 = pow(r44364, r44365);
        double r44367 = r44363 * r44366;
        double r44368 = 0.016666666666666666;
        double r44369 = 5.0;
        double r44370 = pow(r44364, r44369);
        double r44371 = r44368 * r44370;
        double r44372 = 2.0;
        double r44373 = r44372 * r44364;
        double r44374 = r44371 + r44373;
        double r44375 = r44367 + r44374;
        double r44376 = 2.0;
        double r44377 = r44375 / r44376;
        return r44377;
}

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 2019303 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))