Jmat.Real.lambertw, newton loop step

Average Error: 14.0 → 1.6

Time: 27.6s

Precision: 64

• could not determine a ground truth for program body

  1. wj = -4.398408557398186e+133
  2. x = -5.199438090477325e-294

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 3.944304526105059 \cdot 10^{-31}:\\ \;\;\;\;wj \cdot wj + \left(x + \left(x \cdot wj\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{\frac{e^{wj} + wj \cdot e^{wj}}{wj \cdot e^{wj} - x}}\\ \end{array}\]

C

double f(double wj, double x) {
        double r6287485 = wj;
        double r6287486 = exp(r6287485);
        double r6287487 = r6287485 * r6287486;
        double r6287488 = x;
        double r6287489 = r6287487 - r6287488;
        double r6287490 = r6287486 + r6287487;
        double r6287491 = r6287489 / r6287490;
        double r6287492 = r6287485 - r6287491;
        return r6287492;
}

↓

double f(double wj, double x) {
        double r6287493 = wj;
        double r6287494 = exp(r6287493);
        double r6287495 = r6287493 * r6287494;
        double r6287496 = x;
        double r6287497 = r6287495 - r6287496;
        double r6287498 = r6287494 + r6287495;
        double r6287499 = r6287497 / r6287498;
        double r6287500 = r6287493 - r6287499;
        double r6287501 = 3.944304526105059e-31;
        bool r6287502 = r6287500 <= r6287501;
        double r6287503 = r6287493 * r6287493;
        double r6287504 = r6287496 * r6287493;
        double r6287505 = -2.0;
        double r6287506 = r6287504 * r6287505;
        double r6287507 = r6287496 + r6287506;
        double r6287508 = r6287503 + r6287507;
        double r6287509 = 1.0;
        double r6287510 = r6287498 / r6287497;
        double r6287511 = r6287509 / r6287510;
        double r6287512 = r6287493 - r6287511;
        double r6287513 = r6287502 ? r6287508 : r6287512;
        return r6287513;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	14.0
Target	13.4
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 (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 3.944304526105059e-31

   1. Initial program 18.3

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

   2. Taylor expanded around 0 0.5

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

   3. Simplified 0.5

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

   if 3.944304526105059e-31 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

   1. Initial program 3.8

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

   3. Applied *-un-lft-identity 3.8

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

   4. Applied associate-/l* 4.0

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 3.944304526105059 \cdot 10^{-31}:\\ \;\;\;\;wj \cdot wj + \left(x + \left(x \cdot wj\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{\frac{e^{wj} + wj \cdot e^{wj}}{wj \cdot e^{wj} - x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(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))))))