Jmat.Real.lambertw, newton loop step

Average Error: 14.1 → 0.3

Time: 7.1s

Precision: binary64

• could not determine a ground truth for program body

  1. wj = -1.2681344405874179e+38
  2. x = 472694.3100849215

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le -4.810857820983097 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{x}{{wj}^{3} + 1} \cdot \frac{wj \cdot wj + \left(1 - wj \cdot 1\right)}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}\\ \mathbf{elif}\;wj \le 5.72766995050833135 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{wj + 1} \cdot \frac{1}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

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 <= -4.8108578209830974e-09)) {
		VAR = ((double) (((double) (((double) (((double) (x / ((double) (((double) pow(wj, 3.0)) + 1.0)))) * ((double) (((double) (((double) (wj * wj)) + ((double) (1.0 - ((double) (wj * 1.0)))))) / ((double) exp(wj)))))) + wj)) - ((double) (wj / ((double) (wj + 1.0))))));
	} else {
		double VAR_1;
		if ((wj <= 5.727669950508331e-09)) {
			VAR_1 = ((double) (((double) (x + ((double) pow(wj, 2.0)))) - ((double) (2.0 * ((double) (wj * x))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (x / ((double) (wj + 1.0)))) * ((double) (1.0 / ((double) exp(wj)))))) + wj)) - ((double) (wj / ((double) (wj + 1.0))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	14.1
Target	13.1
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if wj < -4.810857820983097e-9

   1. Initial program 3.1

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

   2. Simplified 3.0

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

   4. Applied *-un-lft-identity 3.0

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

   5. Applied flip3-+ 3.2

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

   6. Applied associate-/r/ 3.1

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

   7. Applied times-frac 3.2

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

   8. Simplified 3.2

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

   9. Simplified 3.2

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

   if -4.810857820983097e-9 < wj < 5.72766995050833135e-9

   1. Initial program 13.7

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

   2. Simplified 13.7

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

   3. Taylor expanded around 0 0.2

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

   if 5.72766995050833135e-9 < wj

   1. Initial program 28.5

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

   2. Simplified 2.9

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

   4. Applied div-inv 3.0

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -4.810857820983097 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{x}{{wj}^{3} + 1} \cdot \frac{wj \cdot wj + \left(1 - wj \cdot 1\right)}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}\\ \mathbf{elif}\;wj \le 5.72766995050833135 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{wj + 1} \cdot \frac{1}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020157 
(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))))))