Average Error: 29.4 → 0.3
Time: 11.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00010664185034748744:\\ \;\;\;\;\frac{-1 + e^{x \cdot \left(a \cdot 3\right)}}{1 + \left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \left(\left(e^{a \cdot x} + 1\right) \cdot \sqrt[3]{e^{a \cdot x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(a \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \left(x \cdot \frac{1}{6}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00010664185034748744:\\
\;\;\;\;\frac{-1 + e^{x \cdot \left(a \cdot 3\right)}}{1 + \left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \left(\left(e^{a \cdot x} + 1\right) \cdot \sqrt[3]{e^{a \cdot x}}\right)}\\

\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(a \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \left(x \cdot \frac{1}{6}\right)\\

\end{array}
double f(double a, double x) {
        double r1988990 = a;
        double r1988991 = x;
        double r1988992 = r1988990 * r1988991;
        double r1988993 = exp(r1988992);
        double r1988994 = 1.0;
        double r1988995 = r1988993 - r1988994;
        return r1988995;
}
↓
double f(double a, double x) {
        double r1988996 = a;
        double r1988997 = x;
        double r1988998 = r1988996 * r1988997;
        double r1988999 = -0.00010664185034748744;
        bool r1989000 = r1988998 <= r1988999;
        double r1989001 = -1.0;
        double r1989002 = 3.0;
        double r1989003 = r1988996 * r1989002;
        double r1989004 = r1988997 * r1989003;
        double r1989005 = exp(r1989004);
        double r1989006 = r1989001 + r1989005;
        double r1989007 = 1.0;
        double r1989008 = exp(r1988998);
        double r1989009 = cbrt(r1989008);
        double r1989010 = r1989009 * r1989009;
        double r1989011 = r1989008 + r1989007;
        double r1989012 = r1989011 * r1989009;
        double r1989013 = r1989010 * r1989012;
        double r1989014 = r1989007 + r1989013;
        double r1989015 = r1989006 / r1989014;
        double r1989016 = 0.5;
        double r1989017 = r1988998 * r1988998;
        double r1989018 = r1989016 * r1989017;
        double r1989019 = r1988998 + r1989018;
        double r1989020 = r1988996 * r1989017;
        double r1989021 = 0.16666666666666666;
        double r1989022 = r1988997 * r1989021;
        double r1989023 = r1989020 * r1989022;
        double r1989024 = r1989019 + r1989023;
        double r1989025 = r1989000 ? r1989015 : r1989024;
        return r1989025;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
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) < -0.00010664185034748744

    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 add-cube-cbrt0.0

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

    8. Applied associate-*l*0.0

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

    if -0.00010664185034748744 < (* a x)

    1. Initial program 44.1

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

    2. Taylor expanded around 0 14.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00010664185034748744:\\ \;\;\;\;\frac{-1 + e^{x \cdot \left(a \cdot 3\right)}}{1 + \left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \left(\left(e^{a \cdot x} + 1\right) \cdot \sqrt[3]{e^{a \cdot x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(a \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \left(x \cdot \frac{1}{6}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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))