expq2 (section 3.11)

Average Error: 41.5 → 0.8

Time: 12.0s

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.2568987997710477e+130

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, \frac{1}{12}, \frac{1}{2}\right)\right)\right)\right)}\\ \end{array}\]

C

double f(double x) {
        double r3058453 = x;
        double r3058454 = exp(r3058453);
        double r3058455 = 1.0;
        double r3058456 = r3058454 - r3058455;
        double r3058457 = r3058454 / r3058456;
        return r3058457;
}

↓

double f(double x) {
        double r3058458 = x;
        double r3058459 = exp(r3058458);
        double r3058460 = 0.0;
        bool r3058461 = r3058459 <= r3058460;
        double r3058462 = 1.0;
        double r3058463 = r3058459 - r3058462;
        double r3058464 = exp(r3058463);
        double r3058465 = log(r3058464);
        double r3058466 = r3058459 / r3058465;
        double r3058467 = 1.0;
        double r3058468 = r3058467 / r3058458;
        double r3058469 = 0.08333333333333333;
        double r3058470 = 0.5;
        double r3058471 = fma(r3058458, r3058469, r3058470);
        double r3058472 = log1p(r3058471);
        double r3058473 = expm1(r3058472);
        double r3058474 = log(r3058473);
        double r3058475 = exp(r3058474);
        double r3058476 = r3058468 + r3058475;
        double r3058477 = r3058461 ? r3058466 : r3058476;
        return r3058477;
}

Error

Bits error versus x

Target

Original	41.5
Target	41.2
Herbie	0.8

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

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 add-log-exp 0

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

   4. Applied add-log-exp 0

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

   5. Applied diff-log 0

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

   6. Simplified 0

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

   if 0.0 < (exp x)

   1. Initial program 61.6

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

   2. Taylor expanded around 0 1.1

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

   3. Simplified 1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right) + \frac{1}{x}}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 1.1

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

   7. Applied add-exp-log 1.1

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)} \cdot \color{blue}{e^{\log \left(\sqrt{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)}\right)}} + \frac{1}{x}\]

   8. Applied add-exp-log 1.1

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)}\right)}} \cdot e^{\log \left(\sqrt{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)}\right)} + \frac{1}{x}\]

   9. Applied prod-exp 1.1

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)}\right) + \log \left(\sqrt{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)}\right)}} + \frac{1}{x}\]

   10. Simplified 1.1

       \[\leadsto e^{\color{blue}{\log \left(\mathsf{fma}\left(x, \frac{1}{12}, \frac{1}{2}\right)\right)}} + \frac{1}{x}\]
   11. Using strategy rm

   12. Applied expm1-log1p-u 1.1

       \[\leadsto e^{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, \frac{1}{12}, \frac{1}{2}\right)\right)\right)\right)}} + \frac{1}{x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, \frac{1}{12}, \frac{1}{2}\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))