Average Error: 29.0 → 9.3
Time: 4.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.952525294693658350925354570681857779846 \cdot 10^{-7}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -6.952525294693658350925354570681857779846 \cdot 10^{-7}:\\
\;\;\;\;e^{a \cdot x} - 1\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double a, double x) {
        double r62833 = a;
        double r62834 = x;
        double r62835 = r62833 * r62834;
        double r62836 = exp(r62835);
        double r62837 = 1.0;
        double r62838 = r62836 - r62837;
        return r62838;
}
↓
double f(double a, double x) {
        double r62839 = a;
        double r62840 = x;
        double r62841 = r62839 * r62840;
        double r62842 = -6.952525294693658e-07;
        bool r62843 = r62841 <= r62842;
        double r62844 = exp(r62841);
        double r62845 = 1.0;
        double r62846 = r62844 - r62845;
        double r62847 = 0.5;
        double r62848 = 2.0;
        double r62849 = pow(r62839, r62848);
        double r62850 = pow(r62840, r62848);
        double r62851 = r62849 * r62850;
        double r62852 = 0.16666666666666666;
        double r62853 = 3.0;
        double r62854 = pow(r62839, r62853);
        double r62855 = pow(r62840, r62853);
        double r62856 = r62854 * r62855;
        double r62857 = fma(r62852, r62856, r62841);
        double r62858 = fma(r62847, r62851, r62857);
        double r62859 = r62843 ? r62846 : r62858;
        return r62859;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.0
Target0.2
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -6.952525294693658e-07

    1. Initial program 0.2

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

    if -6.952525294693658e-07 < (* a x)

    1. Initial program 44.5

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

    2. Taylor expanded around 0 14.3

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

    3. Simplified14.3

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.952525294693658350925354570681857779846 \cdot 10^{-7}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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))