Average Error: 29.2 → 0.4
Time: 4.1s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00127281244229751384:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x \cdot a\right)}^{2}}{2} + \mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00127281244229751384:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

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

\end{array}
double f(double a, double x) {
        double r120851 = a;
        double r120852 = x;
        double r120853 = r120851 * r120852;
        double r120854 = exp(r120853);
        double r120855 = 1.0;
        double r120856 = r120854 - r120855;
        return r120856;
}
↓
double f(double a, double x) {
        double r120857 = a;
        double r120858 = x;
        double r120859 = r120857 * r120858;
        double r120860 = -0.0012728124422975138;
        bool r120861 = r120859 <= r120860;
        double r120862 = exp(r120859);
        double r120863 = 1.0;
        double r120864 = r120862 - r120863;
        double r120865 = exp(r120864);
        double r120866 = log(r120865);
        double r120867 = r120858 * r120857;
        double r120868 = 2.0;
        double r120869 = pow(r120867, r120868);
        double r120870 = r120869 / r120868;
        double r120871 = 0.16666666666666666;
        double r120872 = 3.0;
        double r120873 = pow(r120867, r120872);
        double r120874 = fma(r120871, r120873, r120859);
        double r120875 = r120870 + r120874;
        double r120876 = r120861 ? r120866 : r120875;
        return r120876;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.2
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0012728124422975138

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied add-log-exp0.0

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

    5. Applied diff-log0.0

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

    6. Simplified0.0

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

    if -0.0012728124422975138 < (* a x)

    1. Initial program 44.0

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

    2. Taylor expanded around 0 14.7

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

    3. Simplified14.7

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

    5. Applied pow-prod-down8.6

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

    6. Simplified8.6

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

    8. Applied fma-udef8.6

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

    9. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00127281244229751384:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x \cdot a\right)}^{2}}{2} + \mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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))