expq2 (section 3.11)

Average Error: 41.4 → 0.7

Time: 9.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. x = 3.3915171398033327e+295

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.00315224065076235996:\\ \;\;\;\;\frac{1}{\sqrt[3]{1 - \frac{1}{e^{x}}} \cdot \sqrt[3]{1 - \frac{1}{e^{x}}}} \cdot \frac{1}{\sqrt[3]{1 - \frac{1}{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

C

double f(double x) {
        double r107951 = x;
        double r107952 = exp(r107951);
        double r107953 = 1.0;
        double r107954 = r107952 - r107953;
        double r107955 = r107952 / r107954;
        return r107955;
}

↓

double f(double x) {
        double r107956 = x;
        double r107957 = exp(r107956);
        double r107958 = 0.00315224065076236;
        bool r107959 = r107957 <= r107958;
        double r107960 = 1.0;
        double r107961 = 1.0;
        double r107962 = r107961 / r107957;
        double r107963 = r107960 - r107962;
        double r107964 = cbrt(r107963);
        double r107965 = r107964 * r107964;
        double r107966 = r107960 / r107965;
        double r107967 = r107960 / r107964;
        double r107968 = r107966 * r107967;
        double r107969 = 0.5;
        double r107970 = 0.08333333333333333;
        double r107971 = r107970 * r107956;
        double r107972 = r107960 / r107956;
        double r107973 = r107971 + r107972;
        double r107974 = r107969 + r107973;
        double r107975 = r107959 ? r107968 : r107974;
        return r107975;
}

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) < 0.00315224065076236

   1. Initial program 0.0

      \[\frac{e^{x}}{e^{x} - 1}\]
   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-cube-cbrt 0.0

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

   7. Applied add-cube-cbrt 0.0

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

   8. Applied times-frac 0.0

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

   9. Simplified 0.0

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

   10. Simplified 0.0

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

   if 0.00315224065076236 < (exp x)

   1. Initial program 61.7

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

   2. Taylor expanded around 0 1.1

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.00315224065076235996:\\ \;\;\;\;\frac{1}{\sqrt[3]{1 - \frac{1}{e^{x}}} \cdot \sqrt[3]{1 - \frac{1}{e^{x}}}} \cdot \frac{1}{\sqrt[3]{1 - \frac{1}{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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