Average Error: 29.6 → 5.3
Time: 16.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -8.5761719319625954 \cdot 10^{-23}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x + \frac{1}{2} \cdot {a}^{2}\right)\right) + a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -8.5761719319625954 \cdot 10^{-23}:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

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

\end{array}
double f(double a, double x) {
        double r109149 = a;
        double r109150 = x;
        double r109151 = r109149 * r109150;
        double r109152 = exp(r109151);
        double r109153 = 1.0;
        double r109154 = r109152 - r109153;
        return r109154;
}
↓
double f(double a, double x) {
        double r109155 = a;
        double r109156 = x;
        double r109157 = r109155 * r109156;
        double r109158 = -8.576171931962595e-23;
        bool r109159 = r109157 <= r109158;
        double r109160 = exp(r109157);
        double r109161 = 1.0;
        double r109162 = r109160 - r109161;
        double r109163 = exp(r109162);
        double r109164 = log(r109163);
        double r109165 = 0.16666666666666666;
        double r109166 = 3.0;
        double r109167 = pow(r109155, r109166);
        double r109168 = r109165 * r109167;
        double r109169 = r109168 * r109156;
        double r109170 = 0.5;
        double r109171 = 2.0;
        double r109172 = pow(r109155, r109171);
        double r109173 = r109170 * r109172;
        double r109174 = r109169 + r109173;
        double r109175 = r109156 * r109174;
        double r109176 = r109156 * r109175;
        double r109177 = r109176 + r109157;
        double r109178 = r109159 ? r109164 : r109177;
        return r109178;
}

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
Herbie5.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -8.576171931962595e-23

    1. Initial program 2.2

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

    3. Applied add-log-exp2.2

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

    4. Applied add-log-exp2.2

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

    5. Applied diff-log2.3

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

    6. Simplified2.2

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

    if -8.576171931962595e-23 < (* a x)

    1. Initial program 44.1

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

    2. Taylor expanded around 0 13.5

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

    3. Simplified10.4

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

    5. Applied unpow210.4

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

    6. Applied associate-*l*7.0

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -8.5761719319625954 \cdot 10^{-23}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x + \frac{1}{2} \cdot {a}^{2}\right)\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))