expq2 (section 3.11)

Average Error: 41.1 → 0.9

Time: 17.4s

Precision: 64

• could not determine a ground truth for program body

  1. x = 6.77703429302794e+199

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 1.166841640166349666900712269954744631923 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{1 + \sqrt{\frac{1}{e^{x}}}} \cdot \frac{1}{1 - \sqrt{\frac{1}{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

C

double f(double x) {
        double r70527 = x;
        double r70528 = exp(r70527);
        double r70529 = 1.0;
        double r70530 = r70528 - r70529;
        double r70531 = r70528 / r70530;
        return r70531;
}

↓

double f(double x) {
        double r70532 = x;
        double r70533 = exp(r70532);
        double r70534 = 1.1668416401663497e-47;
        bool r70535 = r70533 <= r70534;
        double r70536 = 1.0;
        double r70537 = 1.0;
        double r70538 = r70537 / r70533;
        double r70539 = sqrt(r70538);
        double r70540 = r70536 + r70539;
        double r70541 = r70536 / r70540;
        double r70542 = r70536 - r70539;
        double r70543 = r70536 / r70542;
        double r70544 = r70541 * r70543;
        double r70545 = 0.5;
        double r70546 = 0.08333333333333333;
        double r70547 = r70546 * r70532;
        double r70548 = r70536 / r70532;
        double r70549 = r70547 + r70548;
        double r70550 = r70545 + r70549;
        double r70551 = r70535 ? r70544 : r70550;
        return r70551;
}

Error

Bits error versus x

Target

Original	41.1
Target	40.6
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if (exp x) < 1.1668416401663497e-47

   1. Initial program 0

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

   3. Applied clear-num 0.0

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

   4. Simplified 0.0

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

   6. Applied add-sqr-sqrt 0.0

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

   7. Applied add-sqr-sqrt 0.0

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

   8. Applied difference-of-squares 0.0

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

   9. Applied add-cube-cbrt 0.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt{1} + \sqrt{\frac{1}{e^{x}}}\right) \cdot \left(\sqrt{1} - \sqrt{\frac{1}{e^{x}}}\right)}\]

   10. Applied times-frac 0.0

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{1} + \sqrt{\frac{1}{e^{x}}}} \cdot \frac{\sqrt[3]{1}}{\sqrt{1} - \sqrt{\frac{1}{e^{x}}}}}\]

   11. Simplified 0.0

       \[\leadsto \color{blue}{\frac{1}{1 + \sqrt{\frac{1}{e^{x}}}}} \cdot \frac{\sqrt[3]{1}}{\sqrt{1} - \sqrt{\frac{1}{e^{x}}}}\]

   12. Simplified 0.0

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

   if 1.1668416401663497e-47 < (exp x)

   1. Initial program 61.5

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

   2. Taylor expanded around 0 1.3

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 1.166841640166349666900712269954744631923 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{1 + \sqrt{\frac{1}{e^{x}}}} \cdot \frac{1}{1 - \sqrt{\frac{1}{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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