Jmat.Real.lambertw, newton loop step

Average Error: 13.2 → 0.7

Time: 4.3s

Precision: binary64

• could not determine a ground truth for program body

  1. wj = -3.0804998563194095e+230
  2. x = 6.720218942419971e-139

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le -5.84597177752784035 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \le 1.2051231993133346 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{wj}^{3} + {\left(\frac{\frac{x}{e^{wj}} - wj}{wj + 1}\right)}^{3}}{wj \cdot wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \cdot \left(\frac{\frac{x}{e^{wj}} - wj}{wj + 1} - wj\right)}\\ \end{array}\]

C

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

↓

double code(double wj, double x) {
	double VAR;
	if ((wj <= -5.84597177752784e-09)) {
		VAR = ((double) (wj + ((double) (((double) (((double) (x / ((double) exp(wj)))) - wj)) / ((double) (wj + 1.0))))));
	} else {
		double VAR_1;
		if ((wj <= 1.2051231993133346e-08)) {
			VAR_1 = ((double) (x + ((double) (wj * ((double) (wj + ((double) (x * -2.0))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) pow(wj, 3.0)) + ((double) pow(((double) (((double) (((double) (x / ((double) exp(wj)))) - wj)) / ((double) (wj + 1.0)))), 3.0)))) / ((double) (((double) (wj * wj)) + ((double) (((double) (((double) (((double) (x / ((double) exp(wj)))) - wj)) / ((double) (wj + 1.0)))) * ((double) (((double) (((double) (((double) (x / ((double) exp(wj)))) - wj)) / ((double) (wj + 1.0)))) - wj))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.2
Target	12.6
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 wj < -5.84597177752784035e-9

   1. Initial program 4.3

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

   2. Simplified 4.7

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

   if -5.84597177752784035e-9 < wj < 1.2051231993133346e-8

   1. Initial program 13.1

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

   2. Simplified 13.1

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

   3. Taylor expanded around 0 0.1

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

   4. Simplified 0.2

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

   if 1.2051231993133346e-8 < wj

   1. Initial program 24.2

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

   2. Simplified 3.5

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

   4. Applied flip3-+ 18.0

      \[\leadsto \color{blue}{\frac{{wj}^{3} + {\left(\frac{\frac{x}{e^{wj}} - wj}{wj + 1}\right)}^{3}}{wj \cdot wj + \left(\frac{\frac{x}{e^{wj}} - wj}{wj + 1} \cdot \frac{\frac{x}{e^{wj}} - wj}{wj + 1} - wj \cdot \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\right)}}\]

   5. Simplified 18.0

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -5.84597177752784035 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \le 1.2051231993133346 \cdot 10^{-8}:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{wj}^{3} + {\left(\frac{\frac{x}{e^{wj}} - wj}{wj + 1}\right)}^{3}}{wj \cdot wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \cdot \left(\frac{\frac{x}{e^{wj}} - wj}{wj + 1} - wj\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))))))