expq2 (section 3.11)

Average Error: 39.9 → 0.8

Time: 14.1s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.238106967253489e+148

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 80.66036463545508:\\ \;\;\;\;\frac{\sqrt[3]{e^{x}}}{e^{x} - 1} \cdot \left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{12}} \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt{\frac{1}{12}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

C

double f(double x) {
        double r2784212 = x;
        double r2784213 = exp(r2784212);
        double r2784214 = 1.0;
        double r2784215 = r2784213 - r2784214;
        double r2784216 = r2784213 / r2784215;
        return r2784216;
}

↓

double f(double x) {
        double r2784217 = x;
        double r2784218 = exp(r2784217);
        double r2784219 = 1.0;
        double r2784220 = r2784218 - r2784219;
        double r2784221 = r2784218 / r2784220;
        double r2784222 = 80.66036463545508;
        bool r2784223 = r2784221 <= r2784222;
        double r2784224 = cbrt(r2784218);
        double r2784225 = r2784224 / r2784220;
        double r2784226 = r2784224 * r2784224;
        double r2784227 = r2784225 * r2784226;
        double r2784228 = 0.08333333333333333;
        double r2784229 = sqrt(r2784228);
        double r2784230 = cbrt(r2784217);
        double r2784231 = r2784230 * r2784230;
        double r2784232 = r2784229 * r2784231;
        double r2784233 = r2784230 * r2784232;
        double r2784234 = r2784229 * r2784233;
        double r2784235 = 0.5;
        double r2784236 = r2784219 / r2784217;
        double r2784237 = r2784235 + r2784236;
        double r2784238 = r2784234 + r2784237;
        double r2784239 = r2784223 ? r2784227 : r2784238;
        return r2784239;
}

Error

Bits error versus x

Target

Original	39.9
Target	39.4
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (/ (exp x) (- (exp x) 1)) < 80.66036463545508

   1. Initial program 1.2

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

   3. Applied *-un-lft-identity 1.2

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

   4. Applied add-cube-cbrt 1.2

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

   5. Applied times-frac 1.2

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

   6. Simplified 1.2

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

   if 80.66036463545508 < (/ (exp x) (- (exp x) 1))

   1. Initial program 61.2

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

   2. Taylor expanded around 0 0.6

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 0.6

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{12}} \cdot \sqrt{\frac{1}{12}}\right)} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\]

   5. Applied associate-*l* 0.6

      \[\leadsto \color{blue}{\sqrt{\frac{1}{12}} \cdot \left(\sqrt{\frac{1}{12}} \cdot x\right)} + \left(\frac{1}{x} + \frac{1}{2}\right)\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 0.6

      \[\leadsto \sqrt{\frac{1}{12}} \cdot \left(\sqrt{\frac{1}{12}} \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}\right) + \left(\frac{1}{x} + \frac{1}{2}\right)\]

   8. Applied associate-*r* 0.6

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 80.66036463545508:\\ \;\;\;\;\frac{\sqrt[3]{e^{x}}}{e^{x} - 1} \cdot \left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{12}} \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt{\frac{1}{12}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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