expax (section 3.5)

Average Error: 29.3 → 0.3

Time: 18.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.07174886411196652880040147692852769978344:\\ \;\;\;\;\frac{\log \left(e^{e^{a \cdot \left(3 \cdot x\right)} - 1 \cdot \left(1 \cdot 1\right)}\right)}{\left(1 \cdot e^{a \cdot x} + 1 \cdot 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) + \frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r4987269 = a;
        double r4987270 = x;
        double r4987271 = r4987269 * r4987270;
        double r4987272 = exp(r4987271);
        double r4987273 = 1.0;
        double r4987274 = r4987272 - r4987273;
        return r4987274;
}

↓

double f(double a, double x) {
        double r4987275 = a;
        double r4987276 = x;
        double r4987277 = r4987275 * r4987276;
        double r4987278 = -0.07174886411196653;
        bool r4987279 = r4987277 <= r4987278;
        double r4987280 = 3.0;
        double r4987281 = r4987280 * r4987276;
        double r4987282 = r4987275 * r4987281;
        double r4987283 = exp(r4987282);
        double r4987284 = 1.0;
        double r4987285 = r4987284 * r4987284;
        double r4987286 = r4987284 * r4987285;
        double r4987287 = r4987283 - r4987286;
        double r4987288 = exp(r4987287);
        double r4987289 = log(r4987288);
        double r4987290 = exp(r4987277);
        double r4987291 = r4987284 * r4987290;
        double r4987292 = r4987291 + r4987285;
        double r4987293 = r4987290 * r4987290;
        double r4987294 = r4987292 + r4987293;
        double r4987295 = r4987289 / r4987294;
        double r4987296 = 0.5;
        double r4987297 = r4987296 * r4987277;
        double r4987298 = r4987297 * r4987277;
        double r4987299 = r4987277 + r4987298;
        double r4987300 = 0.16666666666666666;
        double r4987301 = r4987277 * r4987277;
        double r4987302 = r4987277 * r4987301;
        double r4987303 = r4987300 * r4987302;
        double r4987304 = r4987299 + r4987303;
        double r4987305 = r4987279 ? r4987295 : r4987304;
        return r4987305;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.07174886411196653

   1. Initial program 0.0

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

   3. Applied flip3-- 0.0

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

   4. Simplified 0.0

      \[\leadsto \frac{\color{blue}{e^{x \cdot \left(3 \cdot a\right)} - 1 \cdot \left(1 \cdot 1\right)}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}\]
   5. Using strategy rm

   6. Applied add-log-exp 0.0

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

   7. Applied add-log-exp 0.0

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

   8. Applied diff-log 0.0

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

   9. Simplified 0.0

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

   if -0.07174886411196653 < (* a x)

   1. Initial program 44.2

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

   2. Taylor expanded around 0 13.8

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

   3. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.07174886411196652880040147692852769978344:\\ \;\;\;\;\frac{\log \left(e^{e^{a \cdot \left(3 \cdot x\right)} - 1 \cdot \left(1 \cdot 1\right)}\right)}{\left(1 \cdot e^{a \cdot x} + 1 \cdot 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) + \frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))