expax (section 3.5)

Average Error: 29.3 → 0.8

Time: 17.9s

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 r37514 = a;
        double r37515 = x;
        double r37516 = r37514 * r37515;
        double r37517 = exp(r37516);
        double r37518 = 1.0;
        double r37519 = r37517 - r37518;
        return r37519;
}

↓

double f(double a, double x) {
        double r37520 = a;
        double r37521 = x;
        double r37522 = r37520 * r37521;
        double r37523 = -2847.8168594635854;
        bool r37524 = r37522 <= r37523;
        double r37525 = 2.0;
        double r37526 = r37525 * r37522;
        double r37527 = exp(r37526);
        double r37528 = 1.0;
        double r37529 = r37528 * r37528;
        double r37530 = r37527 - r37529;
        double r37531 = exp(r37522);
        double r37532 = r37531 + r37528;
        double r37533 = r37530 / r37532;
        double r37534 = 0.5;
        double r37535 = r37520 * r37522;
        double r37536 = fma(r37534, r37535, r37520);
        double r37537 = r37521 * r37536;
        double r37538 = r37524 ? r37533 : r37537;
        return r37538;
}

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))