Jmat.Real.lambertw, newton loop step

Average Error: 13.7 → 1.6

Time: 32.4s

Precision: 64

• could not determine a ground truth for program body

  1. wj = -2.3127828249285076e+188
  2. x = -1.1962382152566586e+265

Math

\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

↓

\[\begin{array}{l} \mathbf{if}\;wj \le -8.775235746042734 \cdot 10^{-09}:\\ \;\;\;\;wj - \frac{\left(wj - 1\right) \cdot \frac{e^{wj} \cdot wj - x}{wj \cdot wj + -1}}{e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \left(wj + wj\right) \cdot x\right) + wj \cdot wj\\ \end{array}\]

C

double f(double wj, double x) {
        double r9468456 = wj;
        double r9468457 = exp(r9468456);
        double r9468458 = r9468456 * r9468457;
        double r9468459 = x;
        double r9468460 = r9468458 - r9468459;
        double r9468461 = r9468457 + r9468458;
        double r9468462 = r9468460 / r9468461;
        double r9468463 = r9468456 - r9468462;
        return r9468463;
}

↓

double f(double wj, double x) {
        double r9468464 = wj;
        double r9468465 = -8.775235746042734e-09;
        bool r9468466 = r9468464 <= r9468465;
        double r9468467 = 1.0;
        double r9468468 = r9468464 - r9468467;
        double r9468469 = exp(r9468464);
        double r9468470 = r9468469 * r9468464;
        double r9468471 = x;
        double r9468472 = r9468470 - r9468471;
        double r9468473 = r9468464 * r9468464;
        double r9468474 = -1.0;
        double r9468475 = r9468473 + r9468474;
        double r9468476 = r9468472 / r9468475;
        double r9468477 = r9468468 * r9468476;
        double r9468478 = r9468477 / r9468469;
        double r9468479 = r9468464 - r9468478;
        double r9468480 = r9468464 + r9468464;
        double r9468481 = r9468480 * r9468471;
        double r9468482 = r9468471 - r9468481;
        double r9468483 = r9468482 + r9468473;
        double r9468484 = r9468466 ? r9468479 : r9468483;
        return r9468484;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.7
Target	13.0
Herbie	1.6

\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

1. Split input into 2 regimes

2. if wj < -8.775235746042734e-09

   1. Initial program 5.8

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
   2. Using strategy rm

   3. Applied distribute-rgt1-in 5.8

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

   4. Applied associate-/r* 5.8

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{wj + 1}}{e^{wj}}}\]
   5. Using strategy rm

   6. Applied flip-+ 5.9

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

   7. Applied associate-/r/ 6.1

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

   8. Simplified 6.1

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

   if -8.775235746042734e-09 < wj

   1. Initial program 13.9

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

   2. Taylor expanded around 0 1.5

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

   3. Simplified 1.5

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -8.775235746042734 \cdot 10^{-09}:\\ \;\;\;\;wj - \frac{\left(wj - 1\right) \cdot \frac{e^{wj} \cdot wj - x}{wj \cdot wj + -1}}{e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \left(wj + wj\right) \cdot x\right) + wj \cdot wj\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))