Average Error: 28.9 → 14.4
Time: 11.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.8775905144238698 \cdot 10^{82} \lor \neg \left(a \le 6913276300.6773567\right):\\ \;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.8775905144238698 \cdot 10^{82} \lor \neg \left(a \le 6913276300.6773567\right):\\
\;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\

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

\end{array}
double f(double a, double x) {
        double r71958 = a;
        double r71959 = x;
        double r71960 = r71958 * r71959;
        double r71961 = exp(r71960);
        double r71962 = 1.0;
        double r71963 = r71961 - r71962;
        return r71963;
}
↓
double f(double a, double x) {
        double r71964 = a;
        double r71965 = -1.8775905144238698e+82;
        bool r71966 = r71964 <= r71965;
        double r71967 = 6913276300.677357;
        bool r71968 = r71964 <= r71967;
        double r71969 = !r71968;
        bool r71970 = r71966 || r71969;
        double r71971 = 2.0;
        double r71972 = x;
        double r71973 = r71972 * r71964;
        double r71974 = r71971 * r71973;
        double r71975 = exp(r71974);
        double r71976 = 1.0;
        double r71977 = r71976 * r71976;
        double r71978 = r71975 - r71977;
        double r71979 = r71964 * r71972;
        double r71980 = exp(r71979);
        double r71981 = r71980 + r71976;
        double r71982 = r71978 / r71981;
        double r71983 = 0.5;
        double r71984 = pow(r71964, r71971);
        double r71985 = r71983 * r71984;
        double r71986 = 0.16666666666666666;
        double r71987 = 3.0;
        double r71988 = pow(r71964, r71987);
        double r71989 = r71986 * r71988;
        double r71990 = r71989 * r71972;
        double r71991 = r71985 + r71990;
        double r71992 = r71972 * r71991;
        double r71993 = r71964 + r71992;
        double r71994 = r71972 * r71993;
        double r71995 = r71970 ? r71982 : r71994;
        return r71995;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.9
Target0.2
Herbie14.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -1.8775905144238698e+82 or 6913276300.677357 < a

    1. Initial program 20.2

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

    3. Applied flip--20.2

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

    4. Simplified20.1

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

    if -1.8775905144238698e+82 < a < 6913276300.677357

    1. Initial program 33.5

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

    2. Taylor expanded around 0 18.6

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

    3. Simplified11.5

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

  4. Final simplification14.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.8775905144238698 \cdot 10^{82} \lor \neg \left(a \le 6913276300.6773567\right):\\ \;\;\;\;\frac{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))