Average Error: 0.0 → 0.0
Time: 5.9s
Precision: 64
Internal Precision: 128
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[(a \cdot \left(d + b\right) + \left(c \cdot a\right))_*\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original0.0
Target0.0
Herbie0.0
\[a \cdot b + a \cdot \left(c + d\right)\]

Derivation

  1. Initial program 0.0

    \[a \cdot \left(\left(b + c\right) + d\right)\]

  2. Taylor expanded around -inf 0.0

    \[\leadsto \color{blue}{a \cdot d + \left(a \cdot b + a \cdot c\right)}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{(a \cdot \left(d + b\right) + \left(c \cdot a\right))_*}\]

  4. Final simplification0.0

    \[\leadsto (a \cdot \left(d + b\right) + \left(c \cdot a\right))_*\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p14"
  :pre (and (<= 56789 a 98765) (<= 0 b 1) (<= 0 c 0.0016773) (<= 0 d 0.0016773))

  :herbie-target
  (+ (* a b) (* a (+ c d)))

  (* a (+ (+ b c) d)))

Details

Time bar (total: 5.5s)Debug log

sample26.0ms

Algorithm
intervals

simplify47.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

47.0ms
(* a (+ (+ b c) d))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 1 expressions with local error:

0.0b
(* a (+ (+ b c) d))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
1 → 17
Calls

1 calls. Slowest were:

8.0ms
(* a (+ (+ b c) d))

series32.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

32.0ms
(* a (+ (+ b c) d))

simplify419.0ms

Counts
10 → 20
Calls

10 calls. Slowest were:

208.0ms
(+ (* (+ b c) (+ b c)) (- (* d d) (* (+ b c) d)))
86.0ms
(+ (* a d) (+ (* a b) (* a c)))
84.0ms
(+ (* a d) (+ (* a b) (* a c)))

prune149.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize18.0ms

Local error

Found 1 expressions with local error:

0.0b
(fma a (+ d b) (* c a))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(fma a (+ d b) (* c a))

series46.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

46.0ms
(fma a (+ d b) (* c a))

simplify143.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

75.0ms
(+ (* a d) (+ (* a b) (* a c)))
67.0ms
(+ (* a d) (+ (* a b) (* a c)))
1.0ms
0

prune140.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize6.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (sqrt (fma a (+ d b) (* c a))) (sqrt (fma a (+ d b) (* c a))))
0.1b
(sqrt (fma a (+ d b) (* c a)))
0.1b
(sqrt (fma a (+ d b) (* c a)))
0.0b
(fma a (+ d b) (* c a))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

5.0ms
(* (sqrt (fma a (+ d b) (* c a))) (sqrt (fma a (+ d b) (* c a))))
1.0ms
(sqrt (fma a (+ d b) (* c a)))
1.0ms
(sqrt (fma a (+ d b) (* c a)))

series429.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

194.0ms
(sqrt (fma a (+ d b) (* c a)))
168.0ms
(sqrt (fma a (+ d b) (* c a)))
41.0ms
(fma a (+ d b) (* c a))
25.0ms
(* (sqrt (fma a (+ d b) (* c a))) (sqrt (fma a (+ d b) (* c a))))

simplify1.8s

Counts
42 → 82
Calls

42 calls. Slowest were:

370.0ms
(* (* (* (sqrt (fma a (+ d b) (* c a))) (sqrt (fma a (+ d b) (* c a)))) (sqrt (fma a (+ d b) (* c a)))) (* (* (sqrt (fma a (+ d b) (* c a))) (sqrt (fma a (+ d b) (* c a)))) (sqrt (fma a (+ d b) (* c a)))))
117.0ms
(* (fma a (+ d b) (* c a)) (fma a (+ d b) (* c a)))
90.0ms
(sqrt (* (cbrt (fma a (+ d b) (* c a))) (cbrt (fma a (+ d b) (* c a)))))

prune922.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes168.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch2.0ms

end0.0ms

sample1.0s

Algorithm
intervals