expq2 (section 3.11)

Average Error: 40.5 → 0.3

Time: 9.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. x = 6.651092020437287e+111

Math

\[\frac{e^{x}}{e^{x} - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0 \lor \neg \left(e^{x} \le 1.0001616862874831\right):\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

C

double f(double x) {
        double r117424 = x;
        double r117425 = exp(r117424);
        double r117426 = 1.0;
        double r117427 = r117425 - r117426;
        double r117428 = r117425 / r117427;
        return r117428;
}

↓

double f(double x) {
        double r117429 = x;
        double r117430 = exp(r117429);
        double r117431 = 0.0;
        bool r117432 = r117430 <= r117431;
        double r117433 = 1.0001616862874831;
        bool r117434 = r117430 <= r117433;
        double r117435 = !r117434;
        bool r117436 = r117432 || r117435;
        double r117437 = 1.0;
        double r117438 = 1.0;
        double r117439 = r117438 / r117430;
        double r117440 = r117437 - r117439;
        double r117441 = r117437 / r117440;
        double r117442 = 0.5;
        double r117443 = 0.08333333333333333;
        double r117444 = r117443 * r117429;
        double r117445 = r117437 / r117429;
        double r117446 = r117444 + r117445;
        double r117447 = r117442 + r117446;
        double r117448 = r117436 ? r117441 : r117447;
        return r117448;
}

Error

Bits error versus x

Target

Original	40.5
Target	40.1
Herbie	0.3

\[\frac{1}{1 - e^{-x}}\]

Derivation

1. Split input into 2 regimes

2. if (exp x) < 0.0 or 1.0001616862874831 < (exp x)

   1. Initial program 1.3

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

   3. Applied clear-num 1.3

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

   4. Simplified 0.1

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

   if 0.0 < (exp x) < 1.0001616862874831

   1. Initial program 62.0

      \[\frac{e^{x}}{e^{x} - 1}\]

   2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0 \lor \neg \left(e^{x} \le 1.0001616862874831\right):\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))