Average Error: 13.3 → 1.4
Time: 32.6s
Precision: 64
Internal Precision: 128
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 8.047259273015902 \cdot 10^{-23}:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{wj + 1} - \frac{x}{wj \cdot e^{wj} + e^{wj}}\right)\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.3
Target12.6
Herbie1.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 8.047259273015902e-23

    1. Initial program 12.7

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

    2. Taylor expanded around 0 0.9

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

    3. Simplified0.9

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

    if 8.047259273015902e-23 < wj

    1. Initial program 23.0

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    2. Using strategy rm

    3. Applied div-sub23.0

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

    4. Simplified10.9

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 8.047259273015902 \cdot 10^{-23}:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{wj + 1} - \frac{x}{wj \cdot e^{wj} + e^{wj}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))

Details

Time bar (total: 32.0s)Debug log

sample232.0ms

Algorithm
intervals

simplify549.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

549.0ms
(- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 10.8b

localize18.0ms

Local error

Found 4 expressions with local error:

4.4b
(- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))
0.5b
(/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))
0.0b
(+ (exp wj) (* wj (exp wj)))
0.0b
(* wj (exp wj))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

18.0ms
(/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))
7.0ms
(+ (exp wj) (* wj (exp wj)))
2.0ms
(* wj (exp wj))

series169.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

75.0ms
(- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))
35.0ms
(/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))
32.0ms
(+ (exp wj) (* wj (exp wj)))
28.0ms
(* wj (exp wj))

simplify3.4s

Counts
66 → 99
Calls

66 calls. Slowest were:

683.0ms
(/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))
384.0ms
(/ (* wj (exp wj)) (+ (exp wj) (* wj (exp wj))))
345.0ms
(+ (* (exp wj) (exp wj)) (- (* (* wj (exp wj)) (* wj (exp wj))) (* (exp wj) (* wj (exp wj)))))

prune1.0s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize16.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
0 → 0
Calls

0 calls. Slowest were:

series0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

simplify0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

prune17.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.1b

localize30.0ms

Local error

Found 4 expressions with local error:

4.4b
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
0.0b
(/ x (+ (exp wj) (* wj (exp wj))))
0.0b
(+ (exp wj) (* wj (exp wj)))
0.0b
(/ wj (+ 1 wj))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 106
Calls

4 calls. Slowest were:

4.0ms
(+ (exp wj) (* wj (exp wj)))
2.0ms
(/ x (+ (exp wj) (* wj (exp wj))))
1.0ms
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))

series160.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

70.0ms
(/ x (+ (exp wj) (* wj (exp wj))))
43.0ms
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
31.0ms
(+ (exp wj) (* wj (exp wj)))
16.0ms
(/ wj (+ 1 wj))

simplify11.0s

Counts
69 → 118
Calls

69 calls. Slowest were:

484.0ms
(fma (- (+ (* (exp wj) (exp wj)) (- (* (* wj (exp wj)) (* wj (exp wj))) (* (exp wj) (* wj (exp wj)))))) (/ x (+ (pow (exp wj) 3) (pow (* wj (exp wj)) 3))) (* (+ (* (exp wj) (exp wj)) (- (* (* wj (exp wj)) (* wj (exp wj))) (* (exp wj) (* wj (exp wj))))) (/ x (+ (pow (exp wj) 3) (pow (* wj (exp wj)) 3)))))
461.0ms
(fma (- (+ (* (exp wj) (exp wj)) (- (* (* wj (exp wj)) (* wj (exp wj))) (* (exp wj) (* wj (exp wj)))))) (/ x (+ (pow (exp wj) 3) (pow (* wj (exp wj)) 3))) (* (+ (* (exp wj) (exp wj)) (- (* (* wj (exp wj)) (* wj (exp wj))) (* (exp wj) (* wj (exp wj))))) (/ x (+ (pow (exp wj) 3) (pow (* wj (exp wj)) 3)))))
449.0ms
(fma (- (+ (* (exp wj) (exp wj)) (- (* (* wj (exp wj)) (* wj (exp wj))) (* (exp wj) (* wj (exp wj)))))) (/ x (+ (pow (exp wj) 3) (pow (* wj (exp wj)) 3))) (* (+ (* (exp wj) (exp wj)) (- (* (* wj (exp wj)) (* wj (exp wj))) (* (exp wj) (* wj (exp wj))))) (/ x (+ (pow (exp wj) 3) (pow (* wj (exp wj)) 3)))))

prune1.3s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.1b

localize26.0ms

Local error

Found 4 expressions with local error:

4.4b
(- wj (* (exp (- wj)) (/ (- (* wj (exp wj)) x) (+ 1 wj))))
0.5b
(* (exp (- wj)) (/ (- (* wj (exp wj)) x) (+ 1 wj)))
0.0b
(/ (- (* wj (exp wj)) x) (+ 1 wj))
0.0b
(* wj (exp wj))

rewrite44.0ms

Algorithm
rewrite-expression-head
Counts
4 → 98
Calls

4 calls. Slowest were:

23.0ms
(* (exp (- wj)) (/ (- (* wj (exp wj)) x) (+ 1 wj)))
13.0ms
(/ (- (* wj (exp wj)) x) (+ 1 wj))
2.0ms
(- wj (* (exp (- wj)) (/ (- (* wj (exp wj)) x) (+ 1 wj))))

series157.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

60.0ms
(- wj (* (exp (- wj)) (/ (- (* wj (exp wj)) x) (+ 1 wj))))
51.0ms
(* (exp (- wj)) (/ (- (* wj (exp wj)) x) (+ 1 wj)))
30.0ms
(/ (- (* wj (exp wj)) x) (+ 1 wj))
16.0ms
(* wj (exp wj))

simplify3.6s

Counts
74 → 110
Calls

74 calls. Slowest were:

608.0ms
(/ (* (* (- (* wj (exp wj)) x) (- (* wj (exp wj)) x)) (- (* wj (exp wj)) x)) (* (* (+ 1 wj) (+ 1 wj)) (+ 1 wj)))
386.0ms
(* (exp (- wj)) (/ (- (* wj (exp wj)) x) (+ 1 wj)))
240.0ms
(- (+ (* (exp wj) (exp (* -1 wj))) (/ (* x (exp (* -1 wj))) (pow wj 2))) (/ (* x (exp (* -1 wj))) wj))

prune1.1s

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.1b

regimes114.0ms

Accuracy

45% (1.1b remaining)

Error of 1.4b against oracle of 0.3b and baseline of 2.3b

bsearch43.0ms

end0.0ms

sample8.9s

Algorithm
intervals