Average Error: 29.7 → 0.8
Time: 3.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -8268373.4522136636078357696533203125:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left|x \cdot a\right| \cdot \left|x \cdot a\right|, \mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -8268373.4522136636078357696533203125:\\
\;\;\;\;e^{a \cdot x} - 1\\

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

\end{array}
double f(double a, double x) {
        double r118852 = a;
        double r118853 = x;
        double r118854 = r118852 * r118853;
        double r118855 = exp(r118854);
        double r118856 = 1.0;
        double r118857 = r118855 - r118856;
        return r118857;
}
↓
double f(double a, double x) {
        double r118858 = a;
        double r118859 = x;
        double r118860 = r118858 * r118859;
        double r118861 = -8268373.452213664;
        bool r118862 = r118860 <= r118861;
        double r118863 = exp(r118860);
        double r118864 = 1.0;
        double r118865 = r118863 - r118864;
        double r118866 = 0.5;
        double r118867 = r118859 * r118858;
        double r118868 = fabs(r118867);
        double r118869 = r118868 * r118868;
        double r118870 = 0.16666666666666666;
        double r118871 = 3.0;
        double r118872 = pow(r118867, r118871);
        double r118873 = fma(r118870, r118872, r118860);
        double r118874 = fma(r118866, r118869, r118873);
        double r118875 = r118862 ? r118865 : r118874;
        return r118875;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.7
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -8268373.452213664

    1. Initial program 0

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

    if -8268373.452213664 < (* a x)

    1. Initial program 43.9

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

    2. Taylor expanded around 0 15.5

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

    3. Simplified15.5

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

    5. Applied pow-prod-down8.7

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

    6. Simplified8.7

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

    8. Applied add-sqr-sqrt8.7

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

    9. Simplified8.7

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

    10. Simplified1.2

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -8268373.4522136636078357696533203125:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left|x \cdot a\right| \cdot \left|x \cdot a\right|, \mathsf{fma}\left(\frac{1}{6}, {\left(x \cdot a\right)}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +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))