Average Error: 58.0 → 0.7
Time: 4.6s
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 r65360 = x;
        double r65361 = exp(r65360);
        double r65362 = -r65360;
        double r65363 = exp(r65362);
        double r65364 = r65361 - r65363;
        double r65365 = 2.0;
        double r65366 = r65364 / r65365;
        return r65366;
}
↓
double f(double x) {
        double r65367 = 0.3333333333333333;
        double r65368 = x;
        double r65369 = 3.0;
        double r65370 = pow(r65368, r65369);
        double r65371 = r65367 * r65370;
        double r65372 = 0.016666666666666666;
        double r65373 = 5.0;
        double r65374 = pow(r65368, r65373);
        double r65375 = r65372 * r65374;
        double r65376 = r65371 + r65375;
        double r65377 = 2.0;
        double r65378 = r65377 * r65368;
        double r65379 = r65376 + r65378;
        double r65380 = 2.0;
        double r65381 = r65379 / r65380;
        return r65381;
}

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 2019353 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))