Jmat.Real.lambertw, newton loop step

Average Error: 14.2 → 0.6

Time: 5.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t_0 - x}{e^{wj} + t_0} \leq 8.763924776937398 \cdot 10^{-15} \end{array}:\\ \;\;\;\;\mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \mathsf{fma}\left({wj}^{3}, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} \cdot \left(wj + 1\right) - wj \cdot \left(wj + 1\right)}{\left(wj + 1\right) \cdot \left(wj + 1\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (if (let* ((t_0 (* wj (exp wj))))
       (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 8.763924776937398e-15))
   (-
    (fma 2.5 (* x (* wj wj)) (fma wj wj x))
    (fma (pow wj 3.0) (fma x 2.6666666666666665 1.0) (* 2.0 (* wj x))))
   (+
    wj
    (/
     (- (* (/ x (exp wj)) (+ wj 1.0)) (* wj (+ wj 1.0)))
     (* (+ wj 1.0) (+ wj 1.0))))))

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 tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 8.763924776937398e-15) {
		tmp = fma(2.5, (x * (wj * wj)), fma(wj, wj, x)) - fma(pow(wj, 3.0), fma(x, 2.6666666666666665, 1.0), (2.0 * (wj * x)));
	} else {
		tmp = wj + ((((x / exp(wj)) * (wj + 1.0)) - (wj * (wj + 1.0))) / ((wj + 1.0) * (wj + 1.0)));
	}
	return tmp;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	14.2
Target	13.5
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 8.7639247769e-15

   1. Initial program 18.7

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

   2. Simplified 18.7

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

   3. Taylor expanded around 0 0.6

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

   4. Simplified 0.6

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

   5. Taylor expanded around 0 0.6

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

   6. Simplified 0.6

      \[\leadsto \mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \color{blue}{\mathsf{fma}\left({wj}^{3}, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 2 \cdot \left(wj \cdot x\right)\right)} \]

   if 8.7639247769e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.7

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

   2. Simplified 0.4

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

   4. Applied div-sub_binary64 0.4

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

   5. Applied frac-sub_binary64 0.4

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 8.763924776937398 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(2.5, x \cdot \left(wj \cdot wj\right), \mathsf{fma}\left(wj, wj, x\right)\right) - \mathsf{fma}\left({wj}^{3}, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} \cdot \left(wj + 1\right) - wj \cdot \left(wj + 1\right)}{\left(wj + 1\right) \cdot \left(wj + 1\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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))))))