expax (section 3.5)

Average Error: 29.3 → 0.3

Time: 28.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.05104656267540183:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6} + \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) + a \cdot x\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r3796329 = a;
        double r3796330 = x;
        double r3796331 = r3796329 * r3796330;
        double r3796332 = exp(r3796331);
        double r3796333 = 1.0;
        double r3796334 = r3796332 - r3796333;
        return r3796334;
}

↓

double f(double a, double x) {
        double r3796335 = a;
        double r3796336 = x;
        double r3796337 = r3796335 * r3796336;
        double r3796338 = -0.05104656267540183;
        bool r3796339 = r3796337 <= r3796338;
        double r3796340 = exp(r3796337);
        double r3796341 = sqrt(r3796340);
        double r3796342 = 1.0;
        double r3796343 = r3796341 - r3796342;
        double r3796344 = r3796342 + r3796341;
        double r3796345 = r3796343 * r3796344;
        double r3796346 = r3796337 * r3796337;
        double r3796347 = r3796346 * r3796337;
        double r3796348 = 0.16666666666666666;
        double r3796349 = r3796347 * r3796348;
        double r3796350 = 0.5;
        double r3796351 = r3796350 * r3796337;
        double r3796352 = r3796337 * r3796351;
        double r3796353 = r3796352 + r3796337;
        double r3796354 = r3796349 + r3796353;
        double r3796355 = r3796339 ? r3796345 : r3796354;
        return r3796355;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.05104656267540183

   1. Initial program 0.0

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

   3. Applied add-sqr-sqrt 0.0

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

   4. Applied difference-of-sqr-1 0.0

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

   if -0.05104656267540183 < (* a x)

   1. Initial program 44.2

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

   2. Taylor expanded around 0 14.2

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

   3. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.05104656267540183:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6} + \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) + a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :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))