expq2 (section 3.11)

Average Error: 41.4 → 0.7

Time: 5.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.4005802479514868e+144

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9770380934088124247338669192686211317778:\\ \;\;\;\;\frac{e^{x}}{\frac{\frac{{\left(e^{x}\right)}^{6} + \left(-{1}^{6}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

C

double f(double x) {
        double r90838 = x;
        double r90839 = exp(r90838);
        double r90840 = 1.0;
        double r90841 = r90839 - r90840;
        double r90842 = r90839 / r90841;
        return r90842;
}

↓

double f(double x) {
        double r90843 = x;
        double r90844 = exp(r90843);
        double r90845 = 0.9770380934088124;
        bool r90846 = r90844 <= r90845;
        double r90847 = 6.0;
        double r90848 = pow(r90844, r90847);
        double r90849 = 1.0;
        double r90850 = pow(r90849, r90847);
        double r90851 = -r90850;
        double r90852 = r90848 + r90851;
        double r90853 = 3.0;
        double r90854 = pow(r90844, r90853);
        double r90855 = pow(r90849, r90853);
        double r90856 = r90854 + r90855;
        double r90857 = r90852 / r90856;
        double r90858 = r90849 + r90844;
        double r90859 = r90849 * r90858;
        double r90860 = r90843 + r90843;
        double r90861 = exp(r90860);
        double r90862 = r90859 + r90861;
        double r90863 = r90857 / r90862;
        double r90864 = r90844 / r90863;
        double r90865 = 0.5;
        double r90866 = 0.08333333333333333;
        double r90867 = r90866 * r90843;
        double r90868 = 1.0;
        double r90869 = r90868 / r90843;
        double r90870 = r90867 + r90869;
        double r90871 = r90865 + r90870;
        double r90872 = r90846 ? r90864 : r90871;
        return r90872;
}

Error

Bits error versus x

Target

Original	41.4
Target	41.0
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (exp x) < 0.9770380934088124

   1. Initial program 0.0

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

   3. Applied flip3-- 0.0

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

   4. Simplified 0.0

      \[\leadsto \frac{e^{x}}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{\color{blue}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}}}\]
   5. Using strategy rm

   6. Applied flip-- 0.0

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

   7. Simplified 0.0

      \[\leadsto \frac{e^{x}}{\frac{\frac{\color{blue}{{\left(e^{x}\right)}^{6} + \left(-{1}^{6}\right)}}{{\left(e^{x}\right)}^{3} + {1}^{3}}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}}\]

   if 0.9770380934088124 < (exp x)

   1. Initial program 61.6

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

   2. Taylor expanded around 0 1.0

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9770380934088124247338669192686211317778:\\ \;\;\;\;\frac{e^{x}}{\frac{\frac{{\left(e^{x}\right)}^{6} + \left(-{1}^{6}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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