Jmat.Real.lambertw, newton loop step

Average Error: 13.3 → 2.1

Time: 4.3s

Precision: binary64

• could not determine a ground truth for program body

  1. wj = -6.16127684462e+207
  2. x = -6.344854465017469e-11

Math

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↓

\[\]

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) {
	return ((double) (x + ((double) (wj * ((double) (wj - ((double) (x * 2.0))))))));
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.3
Target	12.8
Herbie	2.1

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Derivation

1. Initial program 13.3

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2. Simplified 12.8

   \[\leadsto \]

3. Taylor expanded around 0 2.0

   \[\leadsto \]

4. Simplified 2.1

   \[\leadsto \]

5. Final simplification 2.1

   \[\leadsto \]

Reproduce

herbie shell --seed 2020181 
(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))))))