Average Error: 58.0 → 0.7
Time: 14.5s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{x \cdot \left(4 - \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right)\right)}{2 - \left(x \cdot \frac{1}{3}\right) \cdot x} + \frac{1}{60} \cdot {x}^{5}}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{x \cdot \left(4 - \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right)\right)}{2 - \left(x \cdot \frac{1}{3}\right) \cdot x} + \frac{1}{60} \cdot {x}^{5}}{2}
double f(double x) {
        double r2789347 = x;
        double r2789348 = exp(r2789347);
        double r2789349 = -r2789347;
        double r2789350 = exp(r2789349);
        double r2789351 = r2789348 - r2789350;
        double r2789352 = 2.0;
        double r2789353 = r2789351 / r2789352;
        return r2789353;
}
↓
double f(double x) {
        double r2789354 = x;
        double r2789355 = 4.0;
        double r2789356 = 0.3333333333333333;
        double r2789357 = r2789354 * r2789356;
        double r2789358 = r2789357 * r2789354;
        double r2789359 = r2789358 * r2789358;
        double r2789360 = r2789355 - r2789359;
        double r2789361 = r2789354 * r2789360;
        double r2789362 = 2.0;
        double r2789363 = r2789362 - r2789358;
        double r2789364 = r2789361 / r2789363;
        double r2789365 = 0.016666666666666666;
        double r2789366 = 5.0;
        double r2789367 = pow(r2789354, r2789366);
        double r2789368 = r2789365 * r2789367;
        double r2789369 = r2789364 + r2789368;
        double r2789370 = 2.0;
        double r2789371 = r2789369 / r2789370;
        return r2789371;
}

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

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

  3. Simplified0.7

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

  5. Applied flip-+0.7

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

  6. Applied associate-*r/0.7

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

  7. Simplified0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019169 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))