expq2 (section 3.11)

Average Error: 41.4 → 0.2

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 1.47962565206442419 \cdot 10^{-13} \lor \neg \left(e^{x} \le 1.00035730044387416\right):\\ \;\;\;\;\frac{\frac{1}{1 + \sqrt{\frac{1}{e^{x}}}}}{1 - \sqrt{\frac{1}{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

C

double f(double x) {
        double r93845 = x;
        double r93846 = exp(r93845);
        double r93847 = 1.0;
        double r93848 = r93846 - r93847;
        double r93849 = r93846 / r93848;
        return r93849;
}

↓

double f(double x) {
        double r93850 = x;
        double r93851 = exp(r93850);
        double r93852 = 1.4796256520644242e-13;
        bool r93853 = r93851 <= r93852;
        double r93854 = 1.0003573004438742;
        bool r93855 = r93851 <= r93854;
        double r93856 = !r93855;
        bool r93857 = r93853 || r93856;
        double r93858 = 1.0;
        double r93859 = 1.0;
        double r93860 = r93859 / r93851;
        double r93861 = sqrt(r93860);
        double r93862 = r93858 + r93861;
        double r93863 = r93858 / r93862;
        double r93864 = r93858 - r93861;
        double r93865 = r93863 / r93864;
        double r93866 = 0.08333333333333333;
        double r93867 = r93858 / r93850;
        double r93868 = fma(r93866, r93850, r93867);
        double r93869 = 0.5;
        double r93870 = r93868 + r93869;
        double r93871 = r93857 ? r93865 : r93870;
        return r93871;
}

Error

Bits error versus x

Target

Original	41.4
Target	41.1
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (exp x) < 1.4796256520644242e-13 or 1.0003573004438742 < (exp x)

   1. Initial program 1.2

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

   3. Applied clear-num 1.2

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

   4. Simplified 0.1

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 0.1

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

   7. Applied *-un-lft-identity 0.1

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

   8. Applied difference-of-squares 0.1

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

   9. Applied associate-/r* 0.1

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

   if 1.4796256520644242e-13 < (exp x) < 1.0003573004438742

   1. Initial program 62.2

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

   2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]

   3. Simplified 0.3

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 1.47962565206442419 \cdot 10^{-13} \lor \neg \left(e^{x} \le 1.00035730044387416\right):\\ \;\;\;\;\frac{\frac{1}{1 + \sqrt{\frac{1}{e^{x}}}}}{1 - \sqrt{\frac{1}{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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