expax (section 3.5)

Average Error: 29.6 → 0.8

Time: 16.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.644742988596980756777822009484513188227 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]

C

double f(double a, double x) {
        double r66528 = a;
        double r66529 = x;
        double r66530 = r66528 * r66529;
        double r66531 = exp(r66530);
        double r66532 = 1.0;
        double r66533 = r66531 - r66532;
        return r66533;
}

↓

double f(double a, double x) {
        double r66534 = a;
        double r66535 = x;
        double r66536 = r66534 * r66535;
        double r66537 = -1.6447429885969808e-06;
        bool r66538 = r66536 <= r66537;
        double r66539 = 2.0;
        double r66540 = r66539 * r66536;
        double r66541 = exp(r66540);
        double r66542 = 1.0;
        double r66543 = r66542 * r66542;
        double r66544 = r66541 - r66543;
        double r66545 = exp(r66536);
        double r66546 = r66545 + r66542;
        double r66547 = r66544 / r66546;
        double r66548 = r66535 * r66534;
        double r66549 = 0.16666666666666666;
        double r66550 = 3.0;
        double r66551 = pow(r66536, r66550);
        double r66552 = r66549 * r66551;
        double r66553 = r66548 + r66552;
        double r66554 = r66538 ? r66547 : r66553;
        return r66554;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.6
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) < -1.6447429885969808e-06

   1. Initial program 0.1

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

   3. Applied flip-- 0.1

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

   4. Simplified 0.1

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

   if -1.6447429885969808e-06 < (* a x)

   1. Initial program 44.9

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

   2. Taylor expanded around 0 14.5

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

   3. Simplified 14.5

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

   5. Applied pow-prod-down 4.6

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.644742988596980756777822009484513188227 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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))