Jmat.Real.lambertw, newton loop step

Average Error: 13.8 → 1.1

Time: 4.1s

Precision: binary64

• could not determine a ground truth for program body

  1. wj = -1.5353092605122964e+270
  2. x = 5.740525568479896e+109

Math

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↓

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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) (((double) (((double) (wj * wj)) + ((double) (((double) pow(wj, 4.0)) - ((double) pow(wj, 3.0)))))) + ((double) (((double) (x / ((double) exp(wj)))) / ((double) (wj + 1.0))))));
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.8
Target	13.2
Herbie	1.1

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Derivation

1. Initial program 13.8

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2. Simplified 13.2

   \[\leadsto \]
3. Using strategy rm

4. Applied div-sub 13.2

   \[\leadsto \]

5. Applied associate--r- 7.0

   \[\leadsto \]

6. Taylor expanded around 0 1.1

   \[\leadsto \]

7. Simplified 1.1

   \[\leadsto \]

8. Final simplification 1.1

   \[\leadsto \]

Reproduce

herbie shell --seed 2020180 
(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))))))