Average Error: 13.5 → 1.0
Time: 1.0m
Precision: 64
Internal Precision: 128
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 7.0008525428145445 \cdot 10^{-09}:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{wj + 1} - \frac{x}{wj \cdot e^{wj} + e^{wj}}\right)\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.5
Target12.8
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 7.0008525428145445e-09

    1. Initial program 13.1

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

    2. Taylor expanded around 0 0.8

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

    3. Simplified0.9

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

    if 7.0008525428145445e-09 < wj

    1. Initial program 26.8

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

    3. Applied div-sub26.8

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

    4. Simplified3.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 7.0008525428145445 \cdot 10^{-09}:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{wj + 1} - \frac{x}{wj \cdot e^{wj} + e^{wj}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +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: 1.0m)Debug log

sample205.0ms

Algorithm
intervals

simplify137.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

136.0ms
(- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 15.6b

localize19.0ms

Local error

Found 4 expressions with local error:

4.7b
(- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))
1.0b
(/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))
0.0b
(* wj (exp wj))
0.0b
(* wj (exp wj))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

19.0ms
(/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))
2.0ms
(* wj (exp wj))
2.0ms
(* wj (exp wj))

series139.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

51.0ms
(- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))
46.0ms
(/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))
24.0ms
(* wj (exp wj))
16.0ms
(* wj (exp wj))

simplify3.6s

Counts
63 → 94
Calls

63 calls. Slowest were:

761.0ms
(/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))
440.0ms
(/ (* wj (exp wj)) (+ (exp wj) (* wj (exp wj))))
378.0ms
(+ (* (exp wj) (exp wj)) (- (* (* wj (exp wj)) (* wj (exp wj))) (* (exp wj) (* wj (exp wj)))))

prune954.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.5b

localize11.0ms

Local error

Found 1 expressions with local error:

0.3b
(fma wj (fma x -2 wj) x)

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(fma wj (fma x -2 wj) x)

series17.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

17.0ms
(fma wj (fma x -2 wj) x)

simplify390.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

148.0ms
(- (+ x (pow wj 2)) (* 2 (* x wj)))
141.0ms
(- (+ x (pow wj 2)) (* 2 (* x wj)))
99.0ms
(- (+ (pow wj 2) x) (* 2 (* x wj)))

prune128.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.5b

localize27.0ms

Local error

Found 4 expressions with local error:

4.7b
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
0.0b
(* wj (exp wj))
0.0b
(/ x (+ (exp wj) (* wj (exp wj))))
0.0b
(+ (exp wj) (* wj (exp wj)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 102
Calls

4 calls. Slowest were:

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

series169.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

78.0ms
(/ x (+ (exp wj) (* wj (exp wj))))
45.0ms
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
26.0ms
(* wj (exp wj))
20.0ms
(+ (exp wj) (* wj (exp wj)))

simplify11.2s

Counts
67 → 114
Calls

67 calls. Slowest were:

498.0ms
(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)))))
492.0ms
(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)))))
478.0ms
(fma (- (/ x (+ (exp wj) (* wj (exp wj))))) 1 (* (/ x (+ (exp wj) (* wj (exp wj)))) 1))

prune1.4s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.5b

localize24.0ms

Local error

Found 4 expressions with local error:

4.7b
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
4.7b
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
4.7b
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
0.5b
(cbrt (- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj)))))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 171
Calls

4 calls. Slowest were:

3.0ms
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
3.0ms
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
3.0ms
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))

series252.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

122.0ms
(cbrt (- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj)))))))
54.0ms
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
42.0ms
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
35.0ms
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))

simplify30.4s

Counts
134 → 183
Calls

134 calls. Slowest were:

701.0ms
(fma (- (/ x (+ (exp wj) (* wj (exp wj))))) 1 (* (/ x (+ (exp wj) (* wj (exp wj)))) 1))
550.0ms
(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)))))
543.0ms
(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)))))

prune2.5s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 0.5b

regimes588.0ms

Accuracy

65.6% (0.7b remaining)

Error of 1.0b against oracle of 0.3b and baseline of 2.2b

bsearch64.0ms

end0.0ms

sample8.4s

Algorithm
intervals