Jmat.Real.lambertw, newton loop step

Average Error: 14.1 → 0.7

Time: 5.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj - \frac{t_0 - x}{e^{wj} + t_0}\\ \mathbf{if}\;t_1 \leq -4.766364965098498:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \mathbf{elif}\;t_1 \leq 2.4195265998524308 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot \mathsf{fma}\left(2.5, wj, -2\right), x, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{\mathsf{fma}\left(wj, wj, -1\right)} \cdot \left(wj + -1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))
   (if (<= t_1 -4.766364965098498)
     (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0)))
     (if (<= t_1 2.4195265998524308e-39)
       (fma (* wj (fma 2.5 wj -2.0)) x (fma wj wj x))
       (+ wj (* (/ (- (/ x (exp wj)) wj) (fma wj 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 t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	double tmp;
	if (t_1 <= -4.766364965098498) {
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	} else if (t_1 <= 2.4195265998524308e-39) {
		tmp = fma((wj * fma(2.5, wj, -2.0)), x, fma(wj, wj, x));
	} else {
		tmp = wj + ((((x / exp(wj)) - wj) / fma(wj, wj, -1.0)) * (wj + -1.0));
	}
	return tmp;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	14.1
Target	13.5
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < -4.7663649650984983

   1. Initial program 1.1

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

   2. Simplified 0.1

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

   3. Applied div-inv_binary64 0.1

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

   4. Simplified 0.1

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

   if -4.7663649650984983 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2.41952659985e-39

   1. Initial program 28.1

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

   2. Simplified 28.1

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

   3. Taylor expanded in wj around 0 0.1

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

   4. Simplified 0.1

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

   5. Taylor expanded in wj around 0 0.1

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

   6. Simplified 0.1

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

   if 2.41952659985e-39 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 5.2

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

   2. Simplified 2.1

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

   3. Applied flip-+_binary64 2.2

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

   4. Applied associate-/r/_binary64 2.2

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

   5. Simplified 2.2

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq -4.766364965098498:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \mathbf{elif}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 2.4195265998524308 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot \mathsf{fma}\left(2.5, wj, -2\right), x, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{\mathsf{fma}\left(wj, wj, -1\right)} \cdot \left(wj + -1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))))))