Jmat.Real.lambertw, newton loop step

Average Error: 13.7 → 1.4

Time: 32.9s

Precision: 64

• could not determine a ground truth for program body

  1. wj = 8.335205808559965e+218
  2. x = 1.1248754533781247e-11

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 1.043870210665429517076386450868952238125 \cdot 10^{-17}:\\ \;\;\;\;\left(x + wj \cdot wj\right) - \left(wj + wj\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{1 + wj}\\ \end{array}\]

C

double f(double wj, double x) {
        double r18778657 = wj;
        double r18778658 = exp(r18778657);
        double r18778659 = r18778657 * r18778658;
        double r18778660 = x;
        double r18778661 = r18778659 - r18778660;
        double r18778662 = r18778658 + r18778659;
        double r18778663 = r18778661 / r18778662;
        double r18778664 = r18778657 - r18778663;
        return r18778664;
}

↓

double f(double wj, double x) {
        double r18778665 = wj;
        double r18778666 = exp(r18778665);
        double r18778667 = r18778665 * r18778666;
        double r18778668 = x;
        double r18778669 = r18778667 - r18778668;
        double r18778670 = r18778666 + r18778667;
        double r18778671 = r18778669 / r18778670;
        double r18778672 = r18778665 - r18778671;
        double r18778673 = 1.0438702106654295e-17;
        bool r18778674 = r18778672 <= r18778673;
        double r18778675 = r18778665 * r18778665;
        double r18778676 = r18778668 + r18778675;
        double r18778677 = r18778665 + r18778665;
        double r18778678 = r18778677 * r18778668;
        double r18778679 = r18778676 - r18778678;
        double r18778680 = r18778669 / r18778666;
        double r18778681 = 1.0;
        double r18778682 = r18778681 + r18778665;
        double r18778683 = r18778680 / r18778682;
        double r18778684 = r18778665 - r18778683;
        double r18778685 = r18778674 ? r18778679 : r18778684;
        return r18778685;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.7
Target	13.0
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 1.0438702106654295e-17

   1. Initial program 17.8

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

   2. Taylor expanded around 0 0.8

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

   3. Simplified 0.8

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

   if 1.0438702106654295e-17 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

   1. Initial program 3.1

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

   3. Applied *-un-lft-identity 3.1

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

   4. Applied distribute-rgt-out 3.1

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

   5. Applied associate-/r* 3.1

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{1 + wj}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 1.043870210665429517076386450868952238125 \cdot 10^{-17}:\\ \;\;\;\;\left(x + wj \cdot wj\right) - \left(wj + wj\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{1 + wj}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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