Jmat.Real.lambertw, newton loop step

Average Error: 13.7 → 0.5

Time: 10.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := e^{wj} + t_0\\ \mathbf{if}\;wj - \frac{t_0 - x}{t_1} \leq 9.647393852269467 \cdot 10^{-16}:\\ \;\;\;\;\left(\mathsf{fma}\left(wj, wj, x\right) + x \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(2.5, wj, -2\right), {wj}^{3} \cdot -2.6666666666666665\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{x}{t_1}\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (+ (exp wj) t_0)))
   (if (<= (- wj (/ (- t_0 x) t_1)) 9.647393852269467e-16)
     (-
      (+
       (fma wj wj x)
       (* x (fma wj (fma 2.5 wj -2.0) (* (pow wj 3.0) -2.6666666666666665))))
      (pow wj 3.0))
     (+ (- wj (/ wj (+ wj 1.0))) (/ x t_1)))))

C

double code(double wj, double x) {
	return wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
}

↓

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = exp(wj) + t_0;
	double tmp;
	if ((wj - ((t_0 - x) / t_1)) <= 9.647393852269467e-16) {
		tmp = (fma(wj, wj, x) + (x * fma(wj, fma(2.5, wj, -2.0), (pow(wj, 3.0) * -2.6666666666666665)))) - pow(wj, 3.0);
	} else {
		tmp = (wj - (wj / (wj + 1.0))) + (x / t_1);
	}
	return tmp;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.7
Target	13.0
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 9.647393852e-16

   1. Initial program 17.9

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

   2. Simplified 17.9

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

   3. Taylor expanded in wj around 0 0.4

      \[\leadsto \color{blue}{\left(2.5 \cdot \left({wj}^{2} \cdot x\right) + \left({wj}^{2} + x\right)\right) - \left(2 \cdot \left(wj \cdot x\right) + \left(2.6666666666666665 \cdot \left({wj}^{3} \cdot x\right) + {wj}^{3}\right)\right)} \]

   4. Simplified 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \mathsf{fma}\left(x, \mathsf{fma}\left(2, wj, 2.6666666666666665 \cdot {wj}^{3}\right), {wj}^{3}\right)} \]

   5. Applied fma-udef_binary64 0.4

      \[\leadsto \mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \color{blue}{\left(x \cdot \mathsf{fma}\left(2, wj, 2.6666666666666665 \cdot {wj}^{3}\right) + {wj}^{3}\right)} \]

   6. Applied associate--r+_binary64 0.4

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - x \cdot \mathsf{fma}\left(2, wj, 2.6666666666666665 \cdot {wj}^{3}\right)\right) - {wj}^{3}} \]

   7. Simplified 0.4

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(wj, wj, x\right) + x \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(2.5, wj, -2\right), {wj}^{3} \cdot -2.6666666666666665\right)\right)} - {wj}^{3} \]

   if 9.647393852e-16 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 3.2

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

   2. Applied div-sub_binary64 3.2

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

   3. Applied associate--r-_binary64 3.2

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

   4. Simplified 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 9.647393852269467 \cdot 10^{-16}:\\ \;\;\;\;\left(\mathsf{fma}\left(wj, wj, x\right) + x \cdot \mathsf{fma}\left(wj, \mathsf{fma}\left(2.5, wj, -2\right), {wj}^{3} \cdot -2.6666666666666665\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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