Average Error: 29.1 → 13.5
Time: 17.7s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -5.0289586708105234 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{-1 + e^{\left(x + \left(x + x\right)\right) \cdot \left(3 \cdot a\right)}}{1 + e^{3 \cdot \left(x \cdot a\right)} \cdot \left(e^{3 \cdot \left(x \cdot a\right)} + 1\right)}}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right) + 1}\\ \mathbf{elif}\;a \le 4.1836874807488166 \cdot 10^{+54}:\\ \;\;\;\;x \cdot a + \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\left(x \cdot \frac{1}{6}\right) \cdot a + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + e^{\left(x + \left(x + x\right)\right) \cdot \left(3 \cdot a\right)}}{1 + e^{3 \cdot \left(x \cdot a\right)} \cdot \left(e^{3 \cdot \left(x \cdot a\right)} + 1\right)}}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right) + 1}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \le -5.0289586708105234 \cdot 10^{+104}:\\
\;\;\;\;\frac{\frac{-1 + e^{\left(x + \left(x + x\right)\right) \cdot \left(3 \cdot a\right)}}{1 + e^{3 \cdot \left(x \cdot a\right)} \cdot \left(e^{3 \cdot \left(x \cdot a\right)} + 1\right)}}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right) + 1}\\

\mathbf{elif}\;a \le 4.1836874807488166 \cdot 10^{+54}:\\
\;\;\;\;x \cdot a + \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\left(x \cdot \frac{1}{6}\right) \cdot a + \frac{1}{2}\right)\\

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

\end{array}
double f(double a, double x) {
        double r4180145 = a;
        double r4180146 = x;
        double r4180147 = r4180145 * r4180146;
        double r4180148 = exp(r4180147);
        double r4180149 = 1.0;
        double r4180150 = r4180148 - r4180149;
        return r4180150;
}
↓
double f(double a, double x) {
        double r4180151 = a;
        double r4180152 = -5.0289586708105234e+104;
        bool r4180153 = r4180151 <= r4180152;
        double r4180154 = -1.0;
        double r4180155 = x;
        double r4180156 = r4180155 + r4180155;
        double r4180157 = r4180155 + r4180156;
        double r4180158 = 3.0;
        double r4180159 = r4180158 * r4180151;
        double r4180160 = r4180157 * r4180159;
        double r4180161 = exp(r4180160);
        double r4180162 = r4180154 + r4180161;
        double r4180163 = 1.0;
        double r4180164 = r4180155 * r4180151;
        double r4180165 = r4180158 * r4180164;
        double r4180166 = exp(r4180165);
        double r4180167 = r4180166 + r4180163;
        double r4180168 = r4180166 * r4180167;
        double r4180169 = r4180163 + r4180168;
        double r4180170 = r4180162 / r4180169;
        double r4180171 = exp(r4180164);
        double r4180172 = r4180171 + r4180163;
        double r4180173 = r4180171 * r4180172;
        double r4180174 = r4180173 + r4180163;
        double r4180175 = r4180170 / r4180174;
        double r4180176 = 4.1836874807488166e+54;
        bool r4180177 = r4180151 <= r4180176;
        double r4180178 = r4180164 * r4180164;
        double r4180179 = 0.16666666666666666;
        double r4180180 = r4180155 * r4180179;
        double r4180181 = r4180180 * r4180151;
        double r4180182 = 0.5;
        double r4180183 = r4180181 + r4180182;
        double r4180184 = r4180178 * r4180183;
        double r4180185 = r4180164 + r4180184;
        double r4180186 = r4180177 ? r4180185 : r4180175;
        double r4180187 = r4180153 ? r4180175 : r4180186;
        return r4180187;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.1
Target0.2
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -5.0289586708105234e+104 or 4.1836874807488166e+54 < a

    1. Initial program 16.7

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

    3. Applied flip3--16.8

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

    4. Simplified16.7

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

    5. Simplified16.7

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

    7. Applied flip3--16.7

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

    8. Simplified16.6

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

    9. Simplified16.6

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

    if -5.0289586708105234e+104 < a < 4.1836874807488166e+54

    1. Initial program 34.1

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

    2. Taylor expanded around 0 19.5

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

    3. Simplified12.3

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

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -5.0289586708105234 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{-1 + e^{\left(x + \left(x + x\right)\right) \cdot \left(3 \cdot a\right)}}{1 + e^{3 \cdot \left(x \cdot a\right)} \cdot \left(e^{3 \cdot \left(x \cdot a\right)} + 1\right)}}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right) + 1}\\ \mathbf{elif}\;a \le 4.1836874807488166 \cdot 10^{+54}:\\ \;\;\;\;x \cdot a + \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\left(x \cdot \frac{1}{6}\right) \cdot a + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + e^{\left(x + \left(x + x\right)\right) \cdot \left(3 \cdot a\right)}}{1 + e^{3 \cdot \left(x \cdot a\right)} \cdot \left(e^{3 \cdot \left(x \cdot a\right)} + 1\right)}}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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))