expax (section 3.5)

Average Error: 29.1 → 0.4

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.017454929839710042:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) + \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot x\right) + a \cdot x\\ \end{array}\]

C

double f(double a, double x) {
        double r13487276 = a;
        double r13487277 = x;
        double r13487278 = r13487276 * r13487277;
        double r13487279 = exp(r13487278);
        double r13487280 = 1.0;
        double r13487281 = r13487279 - r13487280;
        return r13487281;
}

↓

double f(double a, double x) {
        double r13487282 = a;
        double r13487283 = x;
        double r13487284 = r13487282 * r13487283;
        double r13487285 = -0.017454929839710042;
        bool r13487286 = r13487284 <= r13487285;
        double r13487287 = exp(r13487284);
        double r13487288 = 1.0;
        double r13487289 = r13487287 - r13487288;
        double r13487290 = 0.5;
        double r13487291 = r13487284 * r13487290;
        double r13487292 = r13487284 * r13487291;
        double r13487293 = 0.16666666666666666;
        double r13487294 = r13487282 * r13487293;
        double r13487295 = r13487284 * r13487284;
        double r13487296 = r13487294 * r13487295;
        double r13487297 = r13487296 * r13487283;
        double r13487298 = r13487292 + r13487297;
        double r13487299 = r13487298 + r13487284;
        double r13487300 = r13487286 ? r13487289 : r13487299;
        return r13487300;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.1
Target	0.1
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.017454929839710042

   1. Initial program 0.0

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

   3. Applied add-log-exp 0.0

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

   4. Applied rem-exp-log 0.0

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

   if -0.017454929839710042 < (* a x)

   1. Initial program 44.3

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

   2. Taylor expanded around 0 14.3

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

   3. Simplified 0.6

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

   5. Applied associate-+l+ 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.017454929839710042:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) + \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot x\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (a x)
  :name "expax (section 3.5)"

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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