Average Error: 58.0 → 0.8
Time: 17.1s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{\left(2 \cdot 2 - \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right)\right) \cdot x}{2 - \left(x \cdot \frac{1}{3}\right) \cdot x} + \frac{1}{60} \cdot {x}^{5}}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{\left(2 \cdot 2 - \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right)\right) \cdot x}{2 - \left(x \cdot \frac{1}{3}\right) \cdot x} + \frac{1}{60} \cdot {x}^{5}}{2}
double f(double x) {
        double r1812313 = x;
        double r1812314 = exp(r1812313);
        double r1812315 = -r1812313;
        double r1812316 = exp(r1812315);
        double r1812317 = r1812314 - r1812316;
        double r1812318 = 2.0;
        double r1812319 = r1812317 / r1812318;
        return r1812319;
}
↓
double f(double x) {
        double r1812320 = 2.0;
        double r1812321 = r1812320 * r1812320;
        double r1812322 = x;
        double r1812323 = 0.3333333333333333;
        double r1812324 = r1812322 * r1812323;
        double r1812325 = r1812324 * r1812322;
        double r1812326 = r1812325 * r1812325;
        double r1812327 = r1812321 - r1812326;
        double r1812328 = r1812327 * r1812322;
        double r1812329 = r1812320 - r1812325;
        double r1812330 = r1812328 / r1812329;
        double r1812331 = 0.016666666666666666;
        double r1812332 = 5.0;
        double r1812333 = pow(r1812322, r1812332);
        double r1812334 = r1812331 * r1812333;
        double r1812335 = r1812330 + r1812334;
        double r1812336 = r1812335 / r1812320;
        return r1812336;
}

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 0.8

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

  3. Simplified0.8

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

  5. Applied flip-+0.8

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

  6. Applied associate-*l/0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2019134 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))