Average Error: 29.6 → 0.4
Time: 16.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.352186635536773:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - 1}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(\frac{1}{6} \cdot x\right) \cdot \left(a \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]
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 + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(\frac{1}{6} \cdot x\right) \cdot \left(a \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\\

\end{array}
double f(double a, double x) {
        double r1725786 = a;
        double r1725787 = x;
        double r1725788 = r1725786 * r1725787;
        double r1725789 = exp(r1725788);
        double r1725790 = 1.0;
        double r1725791 = r1725789 - r1725790;
        return r1725791;
}
↓
double f(double a, double x) {
        double r1725792 = a;
        double r1725793 = x;
        double r1725794 = r1725792 * r1725793;
        double r1725795 = -0.352186635536773;
        bool r1725796 = r1725794 <= r1725795;
        double r1725797 = 3.0;
        double r1725798 = r1725794 * r1725797;
        double r1725799 = exp(r1725798);
        double r1725800 = 1.0;
        double r1725801 = r1725799 - r1725800;
        double r1725802 = exp(r1725794);
        double r1725803 = r1725802 + r1725800;
        double r1725804 = r1725802 * r1725803;
        double r1725805 = r1725804 + r1725800;
        double r1725806 = r1725801 / r1725805;
        double r1725807 = 0.5;
        double r1725808 = r1725794 * r1725794;
        double r1725809 = r1725807 * r1725808;
        double r1725810 = r1725794 + r1725809;
        double r1725811 = 0.16666666666666666;
        double r1725812 = r1725811 * r1725793;
        double r1725813 = r1725792 * r1725808;
        double r1725814 = r1725812 * r1725813;
        double r1725815 = r1725810 + r1725814;
        double r1725816 = r1725796 ? r1725806 : r1725815;
        return r1725816;
}

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

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

    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. Taylor expanded around inf 0.0

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

    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

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

  4. Final simplification0.4

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

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))