Average Error: 3.6 → 0
Time: 3.2s
Precision: 64
Internal Precision: 128
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[2 \cdot \left(\left(d + a\right) + \left(c + b\right)\right)\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.6
Target3.9
Herbie0
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Derivation

  1. Initial program 3.6

    \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
  2. Using strategy rm

  3. Applied associate-+r+2.8

    \[\leadsto \left(a + \color{blue}{\left(\left(b + c\right) + d\right)}\right) \cdot 2\]
  4. Using strategy rm

  5. Applied +-commutative2.8

    \[\leadsto \color{blue}{\left(\left(\left(b + c\right) + d\right) + a\right)} \cdot 2\]
  6. Using strategy rm

  7. Applied associate-+l+0

    \[\leadsto \color{blue}{\left(\left(b + c\right) + \left(d + a\right)\right)} \cdot 2\]

  8. Final simplification0

    \[\leadsto 2 \cdot \left(\left(d + a\right) + \left(c + b\right)\right)\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p6"
  :pre (and (<= -14 a -13) (<= -3 b -2) (<= 3 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2) (* (+ c d) 2))

  (* (+ a (+ b (+ c d))) 2))

Details

Time bar (total: 2.6s)Debug log

sample61.0ms

Algorithm
intervals
Results
256×(pre true 80)
256×(body real 80)

simplify325.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
325.0ms
(* (+ a (+ b (+ c d))) 2)

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 3.6b

localize25.0ms

Local error

Found 2 expressions with local error:

2.7b
(+ a (+ b (+ c d)))
0.4b
(+ b (+ c d))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
2×add-log-exp
2×log1p-expm1-u
2×flip-+
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×flip3-+
2×*-un-lft-identity
2×associate-+r+
2×+-commutative
2×pow1
2×expm1-log1p-u
2×add-sqr-sqrt
Counts
2 → 26
Calls
2 calls:
Slowest
2.0ms
(+ a (+ b (+ c d)))
1.0ms
(+ b (+ c d))

series50.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
32.0ms
(+ a (+ b (+ c d)))
18.0ms
(+ b (+ c d))

simplify61.0ms

Counts
7 → 32
Calls
7 calls:
Slowest
13.0ms
(+ a (+ b c))
12.0ms
(+ b (+ c d))
10.0ms
(+ b (+ c d))
8.0ms
(+ b (+ c d))
6.0ms
(+ b (+ c d))

prune352.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 2.2b

localize19.0ms

Local error

Found 1 expressions with local error:

2.7b
(+ a (+ (+ b c) d))

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×log1p-expm1-u
1×flip-+
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×flip3-+
1×*-un-lft-identity
1×associate-+r+
1×+-commutative
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 13
Calls
1 calls:
Slowest
5.0ms
(+ a (+ (+ b c) d))

series22.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
22.0ms
(+ a (+ (+ b c) d))

simplify34.0ms

Counts
3 → 16
Calls
3 calls:
Slowest
12.0ms
(+ b (+ c d))
12.0ms
(+ b (+ c d))
9.0ms
(+ a (+ b c))

prune147.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 2.0b

localize10.0ms

Local error

Found 1 expressions with local error:

2.7b
(+ (+ (+ b c) d) a)

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
3×fma-def
2×add-cube-cbrt
2×*-un-lft-identity
2×add-sqr-sqrt
1×add-log-exp
1×log1p-expm1-u
1×flip-+
1×add-exp-log
1×add-cbrt-cube
1×flip3-+
1×associate-+l+
1×+-commutative
1×pow1
1×expm1-log1p-u
Counts
1 → 16
Calls
1 calls:
Slowest
14.0ms
(+ (+ (+ b c) d) a)

series27.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
27.0ms
(+ (+ (+ b c) d) a)

simplify41.0ms

Counts
6 → 19
Calls
6 calls:
Slowest
12.0ms
(* (cbrt (+ (+ b c) d)) (cbrt (+ (+ b c) d)))
8.0ms
(+ b (+ c d))
7.0ms
(sqrt (+ (+ b c) d))
6.0ms
(+ a (+ c d))
6.0ms
(+ a (+ c d))

prune179.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize15.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
Counts
0 → 0
Calls
0 calls:
Slowest

series0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

simplify0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

prune8.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample1.2s

Algorithm
intervals
Results
8000×(pre true 80)
8000×(body real 80)