Jmat.Real.lambertw, newton loop step

Average Error: 13.7 → 1.1

Time: 20.9s

Precision: 64

• could not determine a ground truth for program body

  1. wj = 5.0362447883134526e+148
  2. x = 1.0422426701817188e+213

Math

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

↓

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

C

double f(double wj, double x) {
        double r3439941 = wj;
        double r3439942 = exp(r3439941);
        double r3439943 = r3439941 * r3439942;
        double r3439944 = x;
        double r3439945 = r3439943 - r3439944;
        double r3439946 = r3439942 + r3439943;
        double r3439947 = r3439945 / r3439946;
        double r3439948 = r3439941 - r3439947;
        return r3439948;
}

↓

double f(double wj, double x) {
        double r3439949 = x;
        double r3439950 = wj;
        double r3439951 = 1.0;
        double r3439952 = r3439950 + r3439951;
        double r3439953 = r3439949 / r3439952;
        double r3439954 = exp(r3439950);
        double r3439955 = r3439953 / r3439954;
        double r3439956 = r3439950 * r3439950;
        double r3439957 = fma(r3439956, r3439956, r3439956);
        double r3439958 = r3439950 * r3439956;
        double r3439959 = r3439957 - r3439958;
        double r3439960 = r3439955 + r3439959;
        return r3439960;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.7
Target	13.1
Herbie	1.1

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

Derivation

1. Initial program 13.7

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

3. Applied div-sub 13.7

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

4. Applied associate--r- 7.7

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

5. Taylor expanded around 0 1.1

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

6. Simplified 1.1

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

8. Applied distribute-rgt1-in 1.1

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

9. Applied associate-/r* 1.1

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

10. Final simplification 1.1

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(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))))))