Average Error: 29.7 → 0.4
Time: 9.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00565259617492738353:\\ \;\;\;\;\frac{\log \left(e^{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}\right)}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot a, \mathsf{fma}\left(0.5, x \cdot a, 1\right), {\left(a \cdot x\right)}^{3} \cdot 0.16666666666666663\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00565259617492738353:\\
\;\;\;\;\frac{\log \left(e^{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}\right)}{e^{a \cdot x} + 1}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x \cdot a, \mathsf{fma}\left(0.5, x \cdot a, 1\right), {\left(a \cdot x\right)}^{3} \cdot 0.16666666666666663\right)\\

\end{array}
double f(double a, double x) {
        double r140875 = a;
        double r140876 = x;
        double r140877 = r140875 * r140876;
        double r140878 = exp(r140877);
        double r140879 = 1.0;
        double r140880 = r140878 - r140879;
        return r140880;
}
↓
double f(double a, double x) {
        double r140881 = a;
        double r140882 = x;
        double r140883 = r140881 * r140882;
        double r140884 = -0.0056525961749273835;
        bool r140885 = r140883 <= r140884;
        double r140886 = 2.0;
        double r140887 = r140882 * r140881;
        double r140888 = r140886 * r140887;
        double r140889 = exp(r140888);
        double r140890 = 1.0;
        double r140891 = r140890 * r140890;
        double r140892 = r140889 - r140891;
        double r140893 = exp(r140892);
        double r140894 = log(r140893);
        double r140895 = exp(r140883);
        double r140896 = r140895 + r140890;
        double r140897 = r140894 / r140896;
        double r140898 = 0.5;
        double r140899 = fma(r140898, r140887, r140890);
        double r140900 = 3.0;
        double r140901 = pow(r140883, r140900);
        double r140902 = 0.16666666666666663;
        double r140903 = r140901 * r140902;
        double r140904 = fma(r140887, r140899, r140903);
        double r140905 = r140885 ? r140897 : r140904;
        return r140905;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.7
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) < -0.0056525961749273835

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Simplified0.0

      \[\leadsto \frac{\color{blue}{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}}{e^{a \cdot x} + 1}\]
    5. Using strategy rm

    6. Applied add-log-exp0.0

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

    7. Applied add-log-exp0.0

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

    8. Applied diff-log0.0

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

    9. Simplified0.0

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

    if -0.0056525961749273835 < (* a x)

    1. Initial program 44.4

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

    3. Applied flip--44.4

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

    4. Simplified44.4

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

    5. Taylor expanded around 0 14.4

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666663 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)}\]

    6. Simplified0.6

      \[\leadsto \color{blue}{\left(a \cdot x\right) \cdot \left(0.5 \cdot \left(a \cdot x\right) + 1\right) + {\left(a \cdot x\right)}^{3} \cdot 0.16666666666666663}\]

    7. Taylor expanded around inf 14.4

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(1 \cdot \left(a \cdot x\right) + 0.16666666666666663 \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]

    8. Simplified0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot a, \mathsf{fma}\left(0.5, x \cdot a, 1\right), {\left(a \cdot x\right)}^{3} \cdot 0.16666666666666663\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00565259617492738353:\\ \;\;\;\;\frac{\log \left(e^{e^{2 \cdot \left(x \cdot a\right)} - 1 \cdot 1}\right)}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot a, \mathsf{fma}\left(0.5, x \cdot a, 1\right), {\left(a \cdot x\right)}^{3} \cdot 0.16666666666666663\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))