Average Error: 58.1 → 0.4
Time: 4.8s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.0154935820765723566:\\ \;\;\;\;\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-1 \cdot x}}{2}\\ \end{array}\]
\frac{e^{x} - e^{-x}}{2}
↓
\begin{array}{l}
\mathbf{if}\;x \le 0.0154935820765723566:\\
\;\;\;\;\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{x} - e^{-1 \cdot x}}{2}\\

\end{array}
double f(double x) {
        double r130423 = x;
        double r130424 = exp(r130423);
        double r130425 = -r130423;
        double r130426 = exp(r130425);
        double r130427 = r130424 - r130426;
        double r130428 = 2.0;
        double r130429 = r130427 / r130428;
        return r130429;
}
↓
double f(double x) {
        double r130430 = x;
        double r130431 = 0.015493582076572357;
        bool r130432 = r130430 <= r130431;
        double r130433 = 0.3333333333333333;
        double r130434 = 3.0;
        double r130435 = pow(r130430, r130434);
        double r130436 = r130433 * r130435;
        double r130437 = 0.016666666666666666;
        double r130438 = 5.0;
        double r130439 = pow(r130430, r130438);
        double r130440 = r130437 * r130439;
        double r130441 = 2.0;
        double r130442 = r130441 * r130430;
        double r130443 = r130440 + r130442;
        double r130444 = r130436 + r130443;
        double r130445 = 2.0;
        double r130446 = r130444 / r130445;
        double r130447 = exp(r130430);
        double r130448 = -1.0;
        double r130449 = r130448 * r130430;
        double r130450 = exp(r130449);
        double r130451 = r130447 - r130450;
        double r130452 = r130451 / r130445;
        double r130453 = r130432 ? r130446 : r130452;
        return r130453;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 0.015493582076572357

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.4

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

    if 0.015493582076572357 < x

    1. Initial program 0.6

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

    2. Taylor expanded around inf 0.6

      \[\leadsto \frac{\color{blue}{e^{x} - e^{-x}}}{2}\]

    3. Simplified0.6

      \[\leadsto \frac{\color{blue}{e^{x} - e^{-1 \cdot x}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.0154935820765723566:\\ \;\;\;\;\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-1 \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))