exp2 (problem 3.3.7)

Average Error: 29.9 → 0.3

Time: 19.0s

Precision: 64

Math

\[\left(e^{x} - 2\right) + e^{-x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.03372358286109267827557189889375877100974:\\ \;\;\;\;\frac{\left(e^{x} - 2\right) \cdot \left(e^{x} - 2\right) - e^{x \cdot -2}}{\left(e^{x} - 2\right) - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\\ \end{array}\]

C

double f(double x) {
        double r81835 = x;
        double r81836 = exp(r81835);
        double r81837 = 2.0;
        double r81838 = r81836 - r81837;
        double r81839 = -r81835;
        double r81840 = exp(r81839);
        double r81841 = r81838 + r81840;
        return r81841;
}

↓

double f(double x) {
        double r81842 = x;
        double r81843 = -0.03372358286109268;
        bool r81844 = r81842 <= r81843;
        double r81845 = exp(r81842);
        double r81846 = 2.0;
        double r81847 = r81845 - r81846;
        double r81848 = r81847 * r81847;
        double r81849 = -2.0;
        double r81850 = r81842 * r81849;
        double r81851 = exp(r81850);
        double r81852 = r81848 - r81851;
        double r81853 = -r81842;
        double r81854 = exp(r81853);
        double r81855 = r81847 - r81854;
        double r81856 = r81852 / r81855;
        double r81857 = 2.0;
        double r81858 = pow(r81842, r81857);
        double r81859 = 0.002777777777777778;
        double r81860 = 6.0;
        double r81861 = pow(r81842, r81860);
        double r81862 = r81859 * r81861;
        double r81863 = 0.08333333333333333;
        double r81864 = 4.0;
        double r81865 = pow(r81842, r81864);
        double r81866 = r81863 * r81865;
        double r81867 = r81862 + r81866;
        double r81868 = r81858 + r81867;
        double r81869 = r81844 ? r81856 : r81868;
        return r81869;
}

Error

Bits error versus x

Target

Original	29.9
Target	0.0
Herbie	0.3

\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

1. Split input into 2 regimes

2. if x < -0.03372358286109268

   1. Initial program 1.3

      \[\left(e^{x} - 2\right) + e^{-x}\]
   2. Using strategy rm

   3. Applied flip-+ 6.4

      \[\leadsto \color{blue}{\frac{\left(e^{x} - 2\right) \cdot \left(e^{x} - 2\right) - e^{-x} \cdot e^{-x}}{\left(e^{x} - 2\right) - e^{-x}}}\]

   4. Simplified 6.3

      \[\leadsto \frac{\color{blue}{\left(e^{x} - 2\right) \cdot \left(e^{x} - 2\right) - e^{x \cdot -2}}}{\left(e^{x} - 2\right) - e^{-x}}\]

   if -0.03372358286109268 < x

   1. Initial program 30.1

      \[\left(e^{x} - 2\right) + e^{-x}\]

   2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03372358286109267827557189889375877100974:\\ \;\;\;\;\frac{\left(e^{x} - 2\right) \cdot \left(e^{x} - 2\right) - e^{x \cdot -2}}{\left(e^{x} - 2\right) - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))