Average Error: 28.6 → 0.5
Time: 38.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0001298270966635136:\\ \;\;\;\;\frac{e^{\left(a \cdot x + a \cdot x\right) + a \cdot x} + -1}{\left(1 + e^{a \cdot x}\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right) + \left(\frac{1}{6} \cdot \log \left(e^{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)}\right) + a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.0001298270966635136:\\
\;\;\;\;\frac{e^{\left(a \cdot x + a \cdot x\right) + a \cdot x} + -1}{\left(1 + e^{a \cdot x}\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\

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

\end{array}
double f(double a, double x) {
        double r5423128 = a;
        double r5423129 = x;
        double r5423130 = r5423128 * r5423129;
        double r5423131 = exp(r5423130);
        double r5423132 = 1.0;
        double r5423133 = r5423131 - r5423132;
        return r5423133;
}
↓
double f(double a, double x) {
        double r5423134 = a;
        double r5423135 = x;
        double r5423136 = r5423134 * r5423135;
        double r5423137 = -0.0001298270966635136;
        bool r5423138 = r5423136 <= r5423137;
        double r5423139 = r5423136 + r5423136;
        double r5423140 = r5423139 + r5423136;
        double r5423141 = exp(r5423140);
        double r5423142 = -1.0;
        double r5423143 = r5423141 + r5423142;
        double r5423144 = 1.0;
        double r5423145 = exp(r5423136);
        double r5423146 = r5423144 + r5423145;
        double r5423147 = r5423145 * r5423145;
        double r5423148 = r5423146 + r5423147;
        double r5423149 = r5423143 / r5423148;
        double r5423150 = 0.5;
        double r5423151 = r5423136 * r5423150;
        double r5423152 = r5423151 * r5423136;
        double r5423153 = 0.16666666666666666;
        double r5423154 = r5423136 * r5423136;
        double r5423155 = r5423154 * r5423136;
        double r5423156 = exp(r5423155);
        double r5423157 = log(r5423156);
        double r5423158 = r5423153 * r5423157;
        double r5423159 = r5423158 + r5423136;
        double r5423160 = r5423152 + r5423159;
        double r5423161 = r5423138 ? r5423149 : r5423160;
        return r5423161;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.6
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0001298270966635136

    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

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

    5. Simplified0.0

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

    if -0.0001298270966635136 < (* a x)

    1. Initial program 43.6

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

    2. Taylor expanded around 0 14.0

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

    3. Simplified0.6

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

    5. Applied add-log-exp0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0001298270966635136:\\ \;\;\;\;\frac{e^{\left(a \cdot x + a \cdot x\right) + a \cdot x} + -1}{\left(1 + e^{a \cdot x}\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right) + \left(\frac{1}{6} \cdot \log \left(e^{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)}\right) + a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(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))