Average Error: 29.4 → 3.6
Time: 12.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\left(\sqrt[3]{\sqrt{e^{a \cdot x}} - \sqrt{1}} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - \sqrt{1}}\right) \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - \sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, a, \log \left({\left(e^{x \cdot x}\right)}^{\left(\mathsf{fma}\left(\frac{1}{6} \cdot {a}^{3}, x, \frac{1}{2} \cdot {a}^{2}\right)\right)}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\
\;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\left(\sqrt[3]{\sqrt{e^{a \cdot x}} - \sqrt{1}} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - \sqrt{1}}\right) \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - \sqrt{1}}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, a, \log \left({\left(e^{x \cdot x}\right)}^{\left(\mathsf{fma}\left(\frac{1}{6} \cdot {a}^{3}, x, \frac{1}{2} \cdot {a}^{2}\right)\right)}\right)\right)\\

\end{array}
double f(double a, double x) {
        double r73852 = a;
        double r73853 = x;
        double r73854 = r73852 * r73853;
        double r73855 = exp(r73854);
        double r73856 = 1.0;
        double r73857 = r73855 - r73856;
        return r73857;
}
↓
double f(double a, double x) {
        double r73858 = a;
        double r73859 = x;
        double r73860 = r73858 * r73859;
        double r73861 = -0.0009427648001274835;
        bool r73862 = r73860 <= r73861;
        double r73863 = exp(r73860);
        double r73864 = sqrt(r73863);
        double r73865 = 1.0;
        double r73866 = sqrt(r73865);
        double r73867 = r73864 + r73866;
        double r73868 = r73864 - r73866;
        double r73869 = cbrt(r73868);
        double r73870 = r73869 * r73869;
        double r73871 = r73870 * r73869;
        double r73872 = r73867 * r73871;
        double r73873 = r73859 * r73859;
        double r73874 = exp(r73873);
        double r73875 = 0.16666666666666666;
        double r73876 = 3.0;
        double r73877 = pow(r73858, r73876);
        double r73878 = r73875 * r73877;
        double r73879 = 0.5;
        double r73880 = 2.0;
        double r73881 = pow(r73858, r73880);
        double r73882 = r73879 * r73881;
        double r73883 = fma(r73878, r73859, r73882);
        double r73884 = pow(r73874, r73883);
        double r73885 = log(r73884);
        double r73886 = fma(r73859, r73858, r73885);
        double r73887 = r73862 ? r73872 : r73886;
        return r73887;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.4
Target0.2
Herbie3.6
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0009427648001274835

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

      \[\leadsto e^{a \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}\]

    4. Applied add-sqr-sqrt0.0

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - \sqrt{1} \cdot \sqrt{1}\]

    5. Applied difference-of-squares0.0

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

    7. Applied add-cube-cbrt0.0

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

    if -0.0009427648001274835 < (* a x)

    1. Initial program 44.3

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

    2. Taylor expanded around 0 14.2

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

    3. Simplified11.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, a, {x}^{2} \cdot \left(\left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x + \frac{1}{2} \cdot {a}^{2}\right)\right)}\]
    4. Using strategy rm

    5. Applied add-log-exp11.7

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

    6. Simplified5.4

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

  4. Final simplification3.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\left(\sqrt[3]{\sqrt{e^{a \cdot x}} - \sqrt{1}} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - \sqrt{1}}\right) \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - \sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, a, \log \left({\left(e^{x \cdot x}\right)}^{\left(\mathsf{fma}\left(\frac{1}{6} \cdot {a}^{3}, x, \frac{1}{2} \cdot {a}^{2}\right)\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))