Average Error: 13.9 → 2.1
Time: 40.4s
Precision: 64
Internal Precision: 128
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.9
Target13.3
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.9

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

  2. Taylor expanded around 0 2.1

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

  3. Simplified2.1

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

  4. Final simplification2.1

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

Reproduce

herbie shell --seed 2018360 +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: 25.9s)Debug log

start331.0ms

Algorithm
intervals

setup1.0s

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.5b

localize33.0ms

Local error

Found 4 expressions with local error:

4.3b
(- 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))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

13.0ms
(/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))
3.0ms
(+ (exp wj) (* wj (exp wj)))
1.0ms
(- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

series152.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

65.0ms
(- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))
36.0ms
(/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))
27.0ms
(* wj (exp wj))
24.0ms
(+ (exp wj) (* wj (exp wj)))

simplify2.7s

Counts
66 → 99
Calls

66 calls. Slowest were:

305.0ms
(+ (* (exp wj) (exp wj)) (- (* (* wj (exp wj)) (* wj (exp wj))) (* (exp wj) (* wj (exp wj)))))
302.0ms
(/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))
275.0ms
(- (* (* wj (exp wj)) (* wj (exp wj))) (* x x))

prune1.0s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.5b

localize15.0ms

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:

prune28.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.5b

localize16.0ms

Local error

Found 4 expressions with local error:

4.3b
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
0.0b
(+ (exp wj) (* wj (exp wj)))
0.0b
(/ x (+ (exp wj) (* wj (exp wj))))
0.0b
(/ wj (+ 1 wj))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 106
Calls

4 calls. Slowest were:

3.0ms
(+ (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))))))

series140.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

59.0ms
(/ x (+ (exp wj) (* wj (exp wj))))
36.0ms
(- wj (- (/ wj (+ 1 wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
31.0ms
(+ (exp wj) (* wj (exp wj)))
14.0ms
(/ wj (+ 1 wj))

simplify9.0s

Counts
69 → 118
Calls

69 calls. Slowest were:

510.0ms
(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)))))
496.0ms
(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)))))
468.0ms
(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)))))

prune1.5s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.5b

localize11.0ms

Local error

Found 4 expressions with local error:

6.5b
(- wj (- (/ 1 (/ (+ 1 wj) wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
0.2b
(/ 1 (/ (+ 1 wj) wj))
0.0b
(+ (exp wj) (* wj (exp wj)))
0.0b
(/ x (+ (exp wj) (* wj (exp wj))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 103
Calls

4 calls. Slowest were:

7.0ms
(+ (exp wj) (* wj (exp wj)))
3.0ms
(/ x (+ (exp wj) (* wj (exp wj))))
2.0ms
(- wj (- (/ 1 (/ (+ 1 wj) wj)) (/ x (+ (exp wj) (* wj (exp wj))))))

series143.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

66.0ms
(/ x (+ (exp wj) (* wj (exp wj))))
44.0ms
(- wj (- (/ 1 (/ (+ 1 wj) wj)) (/ x (+ (exp wj) (* wj (exp wj))))))
23.0ms
(+ (exp wj) (* wj (exp wj)))
10.0ms
(/ 1 (/ (+ 1 wj) wj))

simplify8.2s

Counts
63 → 115
Calls

63 calls. Slowest were:

539.0ms
(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)))))
493.0ms
(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)))))
450.0ms
(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

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.5b

regimes117.0ms

Accuracy

0% (1.8b remaining)

Error of 2.1b against oracle of 0.3b and baseline of 2.1b

bsearch0.0ms