Average Error: 29.7 → 0.3
Time: 10.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.003862490506531298:\\ \;\;\;\;\frac{\frac{\log \left(e^{e^{3 \cdot \left(\left(a \cdot x\right) \cdot 3\right)} + -1}\right)}{e^{\left(a \cdot x\right) \cdot 3} + \left(e^{\left(a \cdot x\right) \cdot 3} \cdot e^{\left(a \cdot x\right) \cdot 3} + 1\right)}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \left(x \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.003862490506531298:\\
\;\;\;\;\frac{\frac{\log \left(e^{e^{3 \cdot \left(\left(a \cdot x\right) \cdot 3\right)} + -1}\right)}{e^{\left(a \cdot x\right) \cdot 3} + \left(e^{\left(a \cdot x\right) \cdot 3} \cdot e^{\left(a \cdot x\right) \cdot 3} + 1\right)}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1}\\

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

\end{array}
double f(double a, double x) {
        double r2046161 = a;
        double r2046162 = x;
        double r2046163 = r2046161 * r2046162;
        double r2046164 = exp(r2046163);
        double r2046165 = 1.0;
        double r2046166 = r2046164 - r2046165;
        return r2046166;
}
↓
double f(double a, double x) {
        double r2046167 = a;
        double r2046168 = x;
        double r2046169 = r2046167 * r2046168;
        double r2046170 = -0.003862490506531298;
        bool r2046171 = r2046169 <= r2046170;
        double r2046172 = 3.0;
        double r2046173 = r2046169 * r2046172;
        double r2046174 = r2046172 * r2046173;
        double r2046175 = exp(r2046174);
        double r2046176 = -1.0;
        double r2046177 = r2046175 + r2046176;
        double r2046178 = exp(r2046177);
        double r2046179 = log(r2046178);
        double r2046180 = exp(r2046173);
        double r2046181 = r2046180 * r2046180;
        double r2046182 = 1.0;
        double r2046183 = r2046181 + r2046182;
        double r2046184 = r2046180 + r2046183;
        double r2046185 = r2046179 / r2046184;
        double r2046186 = exp(r2046169);
        double r2046187 = r2046186 + r2046182;
        double r2046188 = r2046186 * r2046187;
        double r2046189 = r2046188 + r2046182;
        double r2046190 = r2046185 / r2046189;
        double r2046191 = r2046169 * r2046169;
        double r2046192 = r2046167 * r2046191;
        double r2046193 = 0.16666666666666666;
        double r2046194 = r2046168 * r2046193;
        double r2046195 = r2046192 * r2046194;
        double r2046196 = 0.5;
        double r2046197 = r2046196 * r2046191;
        double r2046198 = r2046197 + r2046169;
        double r2046199 = r2046195 + r2046198;
        double r2046200 = r2046171 ? r2046190 : r2046199;
        return r2046200;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.003862490506531298

    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

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

    5. Simplified0.0

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

    7. Applied flip3-+0.0

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

    8. Simplified0.0

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

    9. Simplified0.0

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

    11. Applied add-log-exp0.0

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

    if -0.003862490506531298 < (* a x)

    1. Initial program 44.1

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

    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. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.003862490506531298:\\ \;\;\;\;\frac{\frac{\log \left(e^{e^{3 \cdot \left(\left(a \cdot x\right) \cdot 3\right)} + -1}\right)}{e^{\left(a \cdot x\right) \cdot 3} + \left(e^{\left(a \cdot x\right) \cdot 3} \cdot e^{\left(a \cdot x\right) \cdot 3} + 1\right)}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \left(x \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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))