Average Error: 29.6 → 0.4
Time: 32.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.352186635536773:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - 1}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\right) + \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.352186635536773:\\
\;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - 1}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1}\\

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

\end{array}
double f(double a, double x) {
        double r3113951 = a;
        double r3113952 = x;
        double r3113953 = r3113951 * r3113952;
        double r3113954 = exp(r3113953);
        double r3113955 = 1.0;
        double r3113956 = r3113954 - r3113955;
        return r3113956;
}
↓
double f(double a, double x) {
        double r3113957 = a;
        double r3113958 = x;
        double r3113959 = r3113957 * r3113958;
        double r3113960 = -0.352186635536773;
        bool r3113961 = r3113959 <= r3113960;
        double r3113962 = 3.0;
        double r3113963 = r3113959 * r3113962;
        double r3113964 = exp(r3113963);
        double r3113965 = 1.0;
        double r3113966 = r3113964 - r3113965;
        double r3113967 = exp(r3113959);
        double r3113968 = r3113967 + r3113965;
        double r3113969 = r3113967 * r3113968;
        double r3113970 = r3113969 + r3113965;
        double r3113971 = r3113966 / r3113970;
        double r3113972 = 0.5;
        double r3113973 = r3113972 * r3113959;
        double r3113974 = r3113959 * r3113973;
        double r3113975 = r3113959 + r3113974;
        double r3113976 = r3113959 * r3113959;
        double r3113977 = r3113976 * r3113959;
        double r3113978 = 0.16666666666666666;
        double r3113979 = r3113977 * r3113978;
        double r3113980 = r3113975 + r3113979;
        double r3113981 = r3113961 ? r3113971 : r3113980;
        return r3113981;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.352186635536773

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

    if -0.352186635536773 < (* a x)

    1. Initial program 44.8

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

    2. Taylor expanded around 0 14.3

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

    3. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.352186635536773:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - 1}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\right) + \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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