Average Error: 58.0 → 0.6
Time: 18.3s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(\frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right) + 2 \cdot x\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{60}}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\left(\frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right) + 2 \cdot x\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{60}}{2}
double f(double x) {
        double r2129346 = x;
        double r2129347 = exp(r2129346);
        double r2129348 = -r2129346;
        double r2129349 = exp(r2129348);
        double r2129350 = r2129347 - r2129349;
        double r2129351 = 2.0;
        double r2129352 = r2129350 / r2129351;
        return r2129352;
}
↓
double f(double x) {
        double r2129353 = 0.3333333333333333;
        double r2129354 = x;
        double r2129355 = r2129354 * r2129354;
        double r2129356 = r2129355 * r2129354;
        double r2129357 = r2129353 * r2129356;
        double r2129358 = 2.0;
        double r2129359 = r2129358 * r2129354;
        double r2129360 = r2129357 + r2129359;
        double r2129361 = r2129355 * r2129356;
        double r2129362 = 0.016666666666666666;
        double r2129363 = r2129361 * r2129362;
        double r2129364 = r2129360 + r2129363;
        double r2129365 = r2129364 / r2129358;
        return r2129365;
}

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 0.6

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

  3. Simplified0.6

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

  4. Taylor expanded around inf 0.6

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

  5. Simplified0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))