Average Error: 29.8 → 9.8
Time: 4.1s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -8.293963492047654929060613942291516367562 \cdot 10^{-20}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -8.293963492047654929060613942291516367562 \cdot 10^{-20}:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\

\end{array}
double f(double a, double x) {
        double r91262 = a;
        double r91263 = x;
        double r91264 = r91262 * r91263;
        double r91265 = exp(r91264);
        double r91266 = 1.0;
        double r91267 = r91265 - r91266;
        return r91267;
}
↓
double f(double a, double x) {
        double r91268 = a;
        double r91269 = x;
        double r91270 = r91268 * r91269;
        double r91271 = -8.293963492047655e-20;
        bool r91272 = r91270 <= r91271;
        double r91273 = exp(r91270);
        double r91274 = 1.0;
        double r91275 = r91273 - r91274;
        double r91276 = exp(r91275);
        double r91277 = log(r91276);
        double r91278 = 0.5;
        double r91279 = 2.0;
        double r91280 = pow(r91268, r91279);
        double r91281 = r91278 * r91280;
        double r91282 = r91281 * r91269;
        double r91283 = r91268 + r91282;
        double r91284 = r91269 * r91283;
        double r91285 = 0.16666666666666666;
        double r91286 = 3.0;
        double r91287 = pow(r91268, r91286);
        double r91288 = pow(r91269, r91286);
        double r91289 = r91287 * r91288;
        double r91290 = r91285 * r91289;
        double r91291 = r91284 + r91290;
        double r91292 = r91272 ? r91277 : r91291;
        return r91292;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.1
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -8.293963492047655e-20

    1. Initial program 1.9

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm

    3. Applied add-log-exp1.9

      \[\leadsto e^{a \cdot x} - \color{blue}{\log \left(e^{1}\right)}\]

    4. Applied add-log-exp1.9

      \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x}}\right)} - \log \left(e^{1}\right)\]

    5. Applied diff-log2.0

      \[\leadsto \color{blue}{\log \left(\frac{e^{e^{a \cdot x}}}{e^{1}}\right)}\]

    6. Simplified1.9

      \[\leadsto \log \color{blue}{\left(e^{e^{a \cdot x} - 1}\right)}\]

    if -8.293963492047655e-20 < (* a x)

    1. Initial program 45.3

      \[e^{a \cdot x} - 1\]

    2. Taylor expanded around 0 14.2

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

    3. Simplified14.2

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -8.293963492047654929060613942291516367562 \cdot 10^{-20}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))