Jmat.Real.lambertw, newton loop step

Average Error: 13.8 → 0.9

Time: 4.1s

Precision: binary64

• could not determine a ground truth for program body

  1. wj = -1.1540550411951562e+29
  2. x = -1.065429437948531e-104

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le 9.2952183473196775 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot e^{\left(0 - \log \left(wj + 1\right)\right) - wj} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

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 <= 9.295218347319678e-09)) {
		VAR = ((double) (((double) (x + ((double) pow(wj, 2.0)))) - ((double) (2.0 * ((double) (wj * x))))));
	} else {
		VAR = ((double) (((double) (((double) (x * ((double) exp(((double) (((double) (0.0 - ((double) log(((double) (wj + 1.0)))))) - wj)))))) + wj)) - ((double) (wj / ((double) (wj + 1.0))))));
	}
	return VAR;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.8
Target	13.1
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if wj < 9.2952183473196775e-9

   1. Initial program 13.4

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

   2. Simplified 13.4

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

   3. Taylor expanded around 0 0.8

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

   if 9.2952183473196775e-9 < wj

   1. Initial program 28.1

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

   2. Simplified 2.7

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

   4. Applied *-un-lft-identity 2.7

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

   5. Applied div-inv 2.7

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

   6. Applied times-frac 2.7

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

   7. Simplified 2.7

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

   9. Applied add-exp-log 2.8

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

   10. Applied rec-exp 2.8

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

   11. Applied div-exp 2.8

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

   12. Simplified 2.8

       \[\leadsto \left(x \cdot e^{\color{blue}{\left(0 - \log \left(wj + 1\right)\right) - wj}} + wj\right) - \frac{wj}{wj + 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.2952183473196775 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot e^{\left(0 - \log \left(wj + 1\right)\right) - wj} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(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))))))