expq2 (section 3.11)

Average Error: 40.1 → 0.8

Time: 15.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = 4.035271246258652e+292

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 17.625799142473483:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{\frac{e^{x} - 1}{\sqrt{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x \cdot \frac{1}{12}} \cdot \sqrt[3]{x \cdot \frac{1}{12}}\right) \cdot \sqrt[3]{x \cdot \frac{1}{12}} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

C

double f(double x) {
        double r1763822 = x;
        double r1763823 = exp(r1763822);
        double r1763824 = 1.0;
        double r1763825 = r1763823 - r1763824;
        double r1763826 = r1763823 / r1763825;
        return r1763826;
}

↓

double f(double x) {
        double r1763827 = x;
        double r1763828 = exp(r1763827);
        double r1763829 = 1.0;
        double r1763830 = r1763828 - r1763829;
        double r1763831 = r1763828 / r1763830;
        double r1763832 = 17.625799142473483;
        bool r1763833 = r1763831 <= r1763832;
        double r1763834 = sqrt(r1763828);
        double r1763835 = r1763830 / r1763834;
        double r1763836 = r1763834 / r1763835;
        double r1763837 = 0.08333333333333333;
        double r1763838 = r1763827 * r1763837;
        double r1763839 = cbrt(r1763838);
        double r1763840 = r1763839 * r1763839;
        double r1763841 = r1763840 * r1763839;
        double r1763842 = r1763829 / r1763827;
        double r1763843 = 0.5;
        double r1763844 = r1763842 + r1763843;
        double r1763845 = r1763841 + r1763844;
        double r1763846 = r1763833 ? r1763836 : r1763845;
        return r1763846;
}

Error

Bits error versus x

Target

Original	40.1
Target	39.7
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (/ (exp x) (- (exp x) 1)) < 17.625799142473483

   1. Initial program 1.2

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

   3. Applied add-sqr-sqrt 1.2

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

   4. Applied associate-/l* 1.2

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

   if 17.625799142473483 < (/ (exp x) (- (exp x) 1))

   1. Initial program 61.1

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

   2. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 0.5

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 17.625799142473483:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{\frac{e^{x} - 1}{\sqrt{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x \cdot \frac{1}{12}} \cdot \sqrt[3]{x \cdot \frac{1}{12}}\right) \cdot \sqrt[3]{x \cdot \frac{1}{12}} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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