Average Error: 38.5 → 0.4
Time: 8.2s
Precision: 64
\[e^{x} - 1\]
\[\begin{array}{l} \mathbf{if}\;x \le -1.446298005470316198547986452638269838644 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x + {x}^{2} \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)\\ \end{array}\]
e^{x} - 1
\begin{array}{l}
\mathbf{if}\;x \le -1.446298005470316198547986452638269838644 \cdot 10^{-4}:\\
\;\;\;\;\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}\\

\mathbf{else}:\\
\;\;\;\;x + {x}^{2} \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)\\

\end{array}
double f(double x) {
        double r102580 = x;
        double r102581 = exp(r102580);
        double r102582 = 1.0;
        double r102583 = r102581 - r102582;
        return r102583;
}

double f(double x) {
        double r102584 = x;
        double r102585 = -0.00014462980054703162;
        bool r102586 = r102584 <= r102585;
        double r102587 = r102584 + r102584;
        double r102588 = exp(r102587);
        double r102589 = 1.0;
        double r102590 = r102589 * r102589;
        double r102591 = r102588 - r102590;
        double r102592 = exp(r102584);
        double r102593 = r102592 + r102589;
        double r102594 = r102591 / r102593;
        double r102595 = 2.0;
        double r102596 = pow(r102584, r102595);
        double r102597 = 0.16666666666666666;
        double r102598 = r102597 * r102584;
        double r102599 = 0.5;
        double r102600 = r102598 + r102599;
        double r102601 = r102596 * r102600;
        double r102602 = r102584 + r102601;
        double r102603 = r102586 ? r102594 : r102602;
        return r102603;
}

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 < -0.00014462980054703162

    1. Initial program 0.1

      \[e^{x} - 1\]
    2. Using strategy rm
    3. Applied flip--0.1

      \[\leadsto \color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}\]
    4. Simplified0.0

      \[\leadsto \frac{\color{blue}{e^{x + x} - 1 \cdot 1}}{e^{x} + 1}\]

    if -0.00014462980054703162 < x

    1. Initial program 58.4

      \[e^{x} - 1\]
    2. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(x + \frac{1}{6} \cdot {x}^{3}\right)}\]
    3. Simplified0.5

      \[\leadsto \color{blue}{x + {x}^{2} \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.
  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.446298005470316198547986452638269838644 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x + {x}^{2} \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 1 
(FPCore (x)
  :name "(exp(x) - 1)"
  :precision binary64
  (- (exp x) 1))