Average Error: 58.1 → 0.6
Time: 14.0s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{\left(8 + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{27}\right) \cdot x}{4 + \left(\left(\left(\frac{1}{3} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{3} \cdot x\right) \cdot x\right) - 2 \cdot \left(\left(\frac{1}{3} \cdot x\right) \cdot x\right)\right)} + {x}^{5} \cdot \frac{1}{60}}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{\left(8 + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{27}\right) \cdot x}{4 + \left(\left(\left(\frac{1}{3} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{3} \cdot x\right) \cdot x\right) - 2 \cdot \left(\left(\frac{1}{3} \cdot x\right) \cdot x\right)\right)} + {x}^{5} \cdot \frac{1}{60}}{2}
double f(double x) {
        double r2607389 = x;
        double r2607390 = exp(r2607389);
        double r2607391 = -r2607389;
        double r2607392 = exp(r2607391);
        double r2607393 = r2607390 - r2607392;
        double r2607394 = 2.0;
        double r2607395 = r2607393 / r2607394;
        return r2607395;
}
↓
double f(double x) {
        double r2607396 = 8.0;
        double r2607397 = x;
        double r2607398 = r2607397 * r2607397;
        double r2607399 = r2607397 * r2607398;
        double r2607400 = r2607399 * r2607399;
        double r2607401 = 0.037037037037037035;
        double r2607402 = r2607400 * r2607401;
        double r2607403 = r2607396 + r2607402;
        double r2607404 = r2607403 * r2607397;
        double r2607405 = 4.0;
        double r2607406 = 0.3333333333333333;
        double r2607407 = r2607406 * r2607397;
        double r2607408 = r2607407 * r2607397;
        double r2607409 = r2607408 * r2607408;
        double r2607410 = 2.0;
        double r2607411 = r2607410 * r2607408;
        double r2607412 = r2607409 - r2607411;
        double r2607413 = r2607405 + r2607412;
        double r2607414 = r2607404 / r2607413;
        double r2607415 = 5.0;
        double r2607416 = pow(r2607397, r2607415);
        double r2607417 = 0.016666666666666666;
        double r2607418 = r2607416 * r2607417;
        double r2607419 = r2607414 + r2607418;
        double r2607420 = 2.0;
        double r2607421 = r2607419 / r2607420;
        return r2607421;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.1

    \[\frac{e^{x} - e^{-x}}{2}\]

  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

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

  5. Applied flip3-+0.6

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

  6. Applied associate-*r/0.6

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

  7. Simplified0.6

    \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(8 + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{27}\right)}}{2 \cdot 2 + \left(\left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) - 2 \cdot \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right)\right)} + {x}^{5} \cdot \frac{1}{60}}{2}\]

  8. Final simplification0.6

    \[\leadsto \frac{\frac{\left(8 + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{27}\right) \cdot x}{4 + \left(\left(\left(\frac{1}{3} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{3} \cdot x\right) \cdot x\right) - 2 \cdot \left(\left(\frac{1}{3} \cdot x\right) \cdot x\right)\right)} + {x}^{5} \cdot \frac{1}{60}}{2}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))