expq2 (section 3.11)

Average Error: 41.4 → 0.2

Time: 3.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r118475 = x;
        double r118476 = exp(r118475);
        double r118477 = 1.0;
        double r118478 = r118476 - r118477;
        double r118479 = r118476 / r118478;
        return r118479;
}

↓

double f(double x) {
        double r118480 = x;
        double r118481 = exp(r118480);
        double r118482 = 1.4796256520644242e-13;
        bool r118483 = r118481 <= r118482;
        double r118484 = 1.0003573004438742;
        bool r118485 = r118481 <= r118484;
        double r118486 = !r118485;
        bool r118487 = r118483 || r118486;
        double r118488 = 1.0;
        double r118489 = 1.0;
        double r118490 = sqrt(r118489);
        double r118491 = sqrt(r118481);
        double r118492 = r118490 / r118491;
        double r118493 = r118488 + r118492;
        double r118494 = r118488 / r118493;
        double r118495 = r118488 - r118492;
        double r118496 = r118494 / r118495;
        double r118497 = 0.5;
        double r118498 = 0.08333333333333333;
        double r118499 = r118498 * r118480;
        double r118500 = r118488 / r118480;
        double r118501 = r118499 + r118500;
        double r118502 = r118497 + r118501;
        double r118503 = r118487 ? r118496 : r118502;
        return r118503;
}

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

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

   7. Applied add-sqr-sqrt 0.1

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

   8. Applied times-frac 0.1

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

   9. Applied *-un-lft-identity 0.1

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

   10. Applied difference-of-squares 0.1

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

   11. Applied associate-/r* 0.1

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

   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. Recombined 2 regimes into one program.

4. Final simplification 0.2

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

Reproduce

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

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

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