Jmat.Real.lambertw, newton loop step

Average Error: 13.8 → 1.0

Time: 19.2s

Precision: 64

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

• could not determine a ground truth for program body

  1. wj = -2.3686796720332175e+206
  2. x = -5.202796025710632e+132

Math

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

↓

\[\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, \sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, -{wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\]

C

double f(double wj, double x) {
        double r220745 = wj;
        double r220746 = exp(r220745);
        double r220747 = r220745 * r220746;
        double r220748 = x;
        double r220749 = r220747 - r220748;
        double r220750 = r220746 + r220747;
        double r220751 = r220749 / r220750;
        double r220752 = r220745 - r220751;
        return r220752;
}

↓

double f(double wj, double x) {
        double r220753 = wj;
        double r220754 = 4.0;
        double r220755 = pow(r220753, r220754);
        double r220756 = fma(r220753, r220753, r220755);
        double r220757 = sqrt(r220756);
        double r220758 = 3.0;
        double r220759 = pow(r220753, r220758);
        double r220760 = -r220759;
        double r220761 = fma(r220757, r220757, r220760);
        double r220762 = x;
        double r220763 = exp(r220753);
        double r220764 = r220762 / r220763;
        double r220765 = 1.0;
        double r220766 = r220765 + r220753;
        double r220767 = r220764 / r220766;
        double r220768 = r220761 + r220767;
        return r220768;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.8
Target	13.3
Herbie	1.0

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

Derivation

1. Initial program 13.8

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

2. Simplified 13.3

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

4. Applied div-sub 13.3

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

5. Applied associate--r- 6.9

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

6. Taylor expanded around 0 1.0

   \[\leadsto \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{1 + wj}\]

7. Simplified 1.0

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(wj, wj, {wj}^{4}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{1 + wj}\]
8. Using strategy rm

9. Applied add-sqr-sqrt 1.0

   \[\leadsto \left(\color{blue}{\sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)} \cdot \sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}} - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\]

10. Applied fma-neg 1.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, \sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, -{wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{1 + wj}\]

11. Final simplification 1.0

    \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, \sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, -{wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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