expax (section 3.5)

Average Error: 29.3 → 0.8

Time: 18.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2847.816859463585387857165187597274780273:\\ \;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{1}{2}, a \cdot \left(a \cdot x\right), a\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r50874 = a;
        double r50875 = x;
        double r50876 = r50874 * r50875;
        double r50877 = exp(r50876);
        double r50878 = 1.0;
        double r50879 = r50877 - r50878;
        return r50879;
}

↓

double f(double a, double x) {
        double r50880 = a;
        double r50881 = x;
        double r50882 = r50880 * r50881;
        double r50883 = -2847.8168594635854;
        bool r50884 = r50882 <= r50883;
        double r50885 = 2.0;
        double r50886 = r50885 * r50882;
        double r50887 = exp(r50886);
        double r50888 = 1.0;
        double r50889 = r50888 * r50888;
        double r50890 = r50887 - r50889;
        double r50891 = exp(r50882);
        double r50892 = r50891 + r50888;
        double r50893 = r50890 / r50892;
        double r50894 = 0.5;
        double r50895 = r50880 * r50882;
        double r50896 = fma(r50894, r50895, r50880);
        double r50897 = r50881 * r50896;
        double r50898 = r50884 ? r50893 : r50897;
        return r50898;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -2847.8168594635854

   1. Initial program 0

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

   3. Applied flip-- 0

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

   4. Simplified 0

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

   if -2847.8168594635854 < (* a x)

   1. Initial program 43.7

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

   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. Simplified 11.3

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

   4. Taylor expanded around 0 8.4

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

   5. Simplified 4.9

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

   7. Applied associate-*l* 1.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2847.816859463585387857165187597274780273:\\ \;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{1}{2}, a \cdot \left(a \cdot x\right), a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.10000000000000001) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))