Average Error: 29.2 → 9.5
Time: 3.4s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.1869555830055673 \cdot 10^{-9}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -1.1869555830055673 \cdot 10^{-9}:\\
\;\;\;\;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 r112727 = a;
        double r112728 = x;
        double r112729 = r112727 * r112728;
        double r112730 = exp(r112729);
        double r112731 = 1.0;
        double r112732 = r112730 - r112731;
        return r112732;
}
↓
double f(double a, double x) {
        double r112733 = a;
        double r112734 = x;
        double r112735 = r112733 * r112734;
        double r112736 = -1.1869555830055673e-09;
        bool r112737 = r112735 <= r112736;
        double r112738 = exp(r112735);
        double r112739 = 1.0;
        double r112740 = r112738 - r112739;
        double r112741 = 0.5;
        double r112742 = 2.0;
        double r112743 = pow(r112733, r112742);
        double r112744 = pow(r112734, r112742);
        double r112745 = r112743 * r112744;
        double r112746 = 0.16666666666666666;
        double r112747 = 3.0;
        double r112748 = pow(r112733, r112747);
        double r112749 = pow(r112734, r112747);
        double r112750 = r112748 * r112749;
        double r112751 = fma(r112746, r112750, r112735);
        double r112752 = fma(r112741, r112745, r112751);
        double r112753 = r112737 ? r112740 : r112752;
        return r112753;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.2
Target0.2
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -1.1869555830055673e-09

    1. Initial program 0.4

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

    if -1.1869555830055673e-09 < (* a x)

    1. Initial program 43.9

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

    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. Simplified14.2

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.1869555830055673 \cdot 10^{-9}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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))