Jmat.Real.lambertw, newton loop step

Average Error: 13.8 → 1.0

Time: 16.9s

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

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

↓

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

C

double f(double wj, double x) {
        double r242487 = wj;
        double r242488 = exp(r242487);
        double r242489 = r242487 * r242488;
        double r242490 = x;
        double r242491 = r242489 - r242490;
        double r242492 = r242488 + r242489;
        double r242493 = r242491 / r242492;
        double r242494 = r242487 - r242493;
        return r242494;
}

↓

double f(double wj, double x) {
        double r242495 = wj;
        double r242496 = 4.0;
        double r242497 = pow(r242495, r242496);
        double r242498 = fma(r242495, r242495, r242497);
        double r242499 = sqrt(r242498);
        double r242500 = 3.0;
        double r242501 = pow(r242495, r242500);
        double r242502 = -r242501;
        double r242503 = fma(r242499, r242499, r242502);
        double r242504 = x;
        double r242505 = exp(r242495);
        double r242506 = r242504 / r242505;
        double r242507 = 1.0;
        double r242508 = r242495 + r242507;
        double r242509 = r242506 / r242508;
        double r242510 = r242503 + r242509;
        return r242510;
}

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

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

4. Applied div-sub 13.3

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

5. Applied associate--r- 6.9

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

6. Taylor expanded around 0 1.0

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

7. Simplified 1.0

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

9. Applied add-sqr-sqrt 1.0

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

10. Applied fma-neg 1.0

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

11. Final simplification 1.0

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

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))))))