expax (section 3.5)

Average Error: 30.4 → 0.8

Time: 17.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.188073158525437477719177204772693645651 \cdot 10^{-7}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]

C

double f(double a, double x) {
        double r64685 = a;
        double r64686 = x;
        double r64687 = r64685 * r64686;
        double r64688 = exp(r64687);
        double r64689 = 1.0;
        double r64690 = r64688 - r64689;
        return r64690;
}

↓

double f(double a, double x) {
        double r64691 = a;
        double r64692 = x;
        double r64693 = r64691 * r64692;
        double r64694 = -1.1880731585254375e-07;
        bool r64695 = r64693 <= r64694;
        double r64696 = exp(r64693);
        double r64697 = 1.0;
        double r64698 = r64696 - r64697;
        double r64699 = exp(r64698);
        double r64700 = log(r64699);
        double r64701 = r64692 * r64691;
        double r64702 = 0.16666666666666666;
        double r64703 = 3.0;
        double r64704 = pow(r64693, r64703);
        double r64705 = r64702 * r64704;
        double r64706 = r64701 + r64705;
        double r64707 = r64695 ? r64700 : r64706;
        return r64707;
}

Error

Bits error versus a

Bits error versus x

Target

Original	30.4
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) < -1.1880731585254375e-07

   1. Initial program 0.2

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

   3. Applied add-log-exp 0.2

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

   4. Applied add-log-exp 0.2

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

   5. Applied diff-log 0.2

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

   6. Simplified 0.2

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

   if -1.1880731585254375e-07 < (* a x)

   1. Initial program 45.2

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

   2. Taylor expanded around 0 14.7

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

   3. Simplified 14.7

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

   5. Applied pow-prod-down 4.5

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

   6. Taylor expanded around 0 1.1

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

   7. Simplified 1.1

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.188073158525437477719177204772693645651 \cdot 10^{-7}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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))