expq2 (section 3.11)

Average Error: 41.3 → 1.0

Time: 20.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = 5.286892203419643e+22

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{{\left(\frac{\sqrt{1}}{\sqrt{e^{x}}}\right)}^{3}}} \cdot \frac{1}{1 - \frac{\sqrt{1}}{\sqrt{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

C

double f(double x) {
        double r72814 = x;
        double r72815 = exp(r72814);
        double r72816 = 1.0;
        double r72817 = r72815 - r72816;
        double r72818 = r72815 / r72817;
        return r72818;
}

↓

double f(double x) {
        double r72819 = x;
        double r72820 = exp(r72819);
        double r72821 = 0.0;
        bool r72822 = r72820 <= r72821;
        double r72823 = 1.0;
        double r72824 = 1.0;
        double r72825 = sqrt(r72824);
        double r72826 = sqrt(r72820);
        double r72827 = r72825 / r72826;
        double r72828 = 3.0;
        double r72829 = pow(r72827, r72828);
        double r72830 = cbrt(r72829);
        double r72831 = r72823 + r72830;
        double r72832 = r72823 / r72831;
        double r72833 = r72823 - r72827;
        double r72834 = r72823 / r72833;
        double r72835 = r72832 * r72834;
        double r72836 = 0.5;
        double r72837 = 0.08333333333333333;
        double r72838 = r72837 * r72819;
        double r72839 = r72823 / r72819;
        double r72840 = r72838 + r72839;
        double r72841 = r72836 + r72840;
        double r72842 = r72822 ? r72835 : r72841;
        return r72842;
}

Error

Bits error versus x

Target

Original	41.3
Target	40.8
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if (exp x) < 0.0

   1. Initial program 0

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

   3. Applied clear-num 0

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

   4. Simplified 0

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

   6. Applied add-sqr-sqrt 0

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

   7. Applied add-sqr-sqrt 0

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

   8. Applied times-frac 0

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

   9. Applied add-sqr-sqrt 0

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

   10. Applied difference-of-squares 0

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

   11. Applied add-cube-cbrt 0

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

   12. Applied times-frac 0

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

   13. Simplified 0

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

   14. Simplified 0

       \[\leadsto \frac{1}{1 + \frac{\sqrt{1}}{\sqrt{e^{x}}}} \cdot \color{blue}{\frac{1}{1 - \frac{\sqrt{1}}{\sqrt{e^{x}}}}}\]
   15. Using strategy rm

   16. Applied add-cbrt-cube 0

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

   17. Applied add-cbrt-cube 0

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

   18. Applied cbrt-undiv 0

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

   19. Simplified 0

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

   if 0.0 < (exp x)

   1. Initial program 61.1

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

   2. Taylor expanded around 0 1.4

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{{\left(\frac{\sqrt{1}}{\sqrt{e^{x}}}\right)}^{3}}} \cdot \frac{1}{1 - \frac{\sqrt{1}}{\sqrt{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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