Average Error: 0.0 → 0.0
Time: 6.9s
Precision: 64
Internal Precision: 128
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[\left(b \cdot a + c \cdot a\right) + d \cdot a\]

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

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. Using strategy rm

  3. Applied distribute-lft-in0.0

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

  5. Applied distribute-rgt-in0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019016 
(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: 6.2s)Debug log

sample54.0ms

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

simplify26.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
26.0ms
(* a (+ (+ b c) d))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize12.0ms

Local error

Found 2 expressions with local error:

0.1b
(* a (+ (+ b c) d))
0.0b
(+ (+ b c) d)

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
3×add-cube-cbrt
3×associate-*r*
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
2×flip-+
2×add-exp-log
2×associate-*r/
2×add-cbrt-cube
2×flip3-+
2×pow1
1×*-commutative
1×distribute-lft-in
1×associate-+l+
1×+-commutative
1×distribute-rgt-in
Counts
2 → 26
Calls
2 calls:
Slowest
8.0ms
(* a (+ (+ b c) d))
5.0ms
(+ (+ b c) d)

series64.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
45.0ms
(* a (+ (+ b c) d))
19.0ms
(+ (+ b c) d)

simplify323.0ms

Counts
13 → 32
Calls
13 calls:
Slowest
165.0ms
(+ (* (+ b c) (+ b c)) (- (* d d) (* (+ b c) d)))
48.0ms
(+ (* a d) (+ (* a b) (* a c)))
28.0ms
(+ (* a d) (+ (* a b) (* a c)))
14.0ms
(- (+ b c) d)
14.0ms
(cbrt (+ (+ b c) d))

prune330.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 2 expressions with local error:

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

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
3×add-cube-cbrt
3×associate-*r*
3×*-un-lft-identity
3×add-sqr-sqrt
2×flip-+
2×add-exp-log
2×associate-*r/
2×add-cbrt-cube
2×flip3-+
2×distribute-lft-in
2×associate-+l+
2×pow1
2×distribute-rgt-in
1×distribute-lft-out
1×*-commutative
1×sum-log
1×+-commutative
Counts
2 → 29
Calls
2 calls:
Slowest
3.0ms
(+ (* a (+ b c)) (* a d))
1.0ms
(* a (+ b c))

series49.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
26.0ms
(+ (* a (+ b c)) (* a d))
23.0ms
(* a (+ b c))

simplify281.0ms

Counts
17 → 35
Calls
17 calls:
Slowest
98.0ms
(* (exp (* a (+ b c))) (exp (* a d)))
61.0ms
(+ (* b b) (- (* c c) (* b c)))
51.0ms
(+ (* a d) (+ (* a b) (* a c)))
27.0ms
(+ (* a d) (+ (* a b) (* a c)))
10.0ms
(+ (* a b) (* a c))

prune372.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0b

localize23.0ms

Local error

Found 2 expressions with local error:

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Rules
9×add-log-exp
4×sum-log
2×flip-+
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×distribute-rgt-out
2×flip3-+
2×*-un-lft-identity
2×+-commutative
2×pow1
2×add-sqr-sqrt
1×distribute-lft-out
1×associate-+l+
Counts
2 → 26
Calls
2 calls:
Slowest
16.0ms
(+ (+ (* b a) (* c a)) (* a d))
5.0ms
(+ (* b a) (* c a))

series46.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
27.0ms
(+ (+ (* b a) (* c a)) (* a d))
18.0ms
(+ (* b a) (* c a))

simplify279.0ms

Counts
12 → 32
Calls
12 calls:
Slowest
98.0ms
(* (exp (+ (* b a) (* c a))) (exp (* a d)))
62.0ms
(* (* (exp (* b a)) (exp (* c a))) (exp (* a d)))
47.0ms
(+ (* a d) (+ (* a b) (* a c)))
26.0ms
(+ (* a d) (+ (* a b) (* a c)))
12.0ms
(* (exp (* b a)) (exp (* c a)))

prune342.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (* a (sqrt (+ b c))) (sqrt (+ b c)))
0.2b
(* a (sqrt (+ b c)))
0.0b
(+ (* (* a (sqrt (+ b c))) (sqrt (+ b c))) (* a d))
0.0b
(sqrt (+ b c))

rewrite34.0ms

Algorithm
rewrite-expression-head
Rules
16×sqrt-div
12×associate-*r*
10×associate-*r/
10×add-sqr-sqrt
9×flip-+
9×sqrt-prod
9×add-cube-cbrt
9×flip3-+
9×*-un-lft-identity
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
6×pow1
4×frac-times
2×*-commutative
2×associate-*l/
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×pow1/2
1×sum-log
1×rem-sqrt-square
1×+-commutative
Counts
4 → 67
Calls
4 calls:
Slowest
16.0ms
(+ (* (* a (sqrt (+ b c))) (sqrt (+ b c))) (* a d))
13.0ms
(* (* a (sqrt (+ b c))) (sqrt (+ b c)))
3.0ms
(* a (sqrt (+ b c)))
1.0ms
(sqrt (+ b c))

series146.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
44.0ms
(+ (* (* a (sqrt (+ b c))) (sqrt (+ b c))) (* a d))
38.0ms
(* a (sqrt (+ b c)))
35.0ms
(* (* a (sqrt (+ b c))) (sqrt (+ b c)))
29.0ms
(sqrt (+ b c))

simplify1.7s

Counts
48 → 79
Calls
48 calls:
Slowest
613.0ms
(* (* (* (* a (sqrt (+ b c))) (* a (sqrt (+ b c)))) (* a (sqrt (+ b c)))) (* (* (sqrt (+ b c)) (sqrt (+ b c))) (sqrt (+ b c))))
189.0ms
(* (* a (sqrt (- (* b b) (* c c)))) (sqrt (- (* b b) (* c c))))
86.0ms
(* (* a (sqrt (- (* b b) (* c c)))) (sqrt (+ b c)))
84.0ms
(* (* a (sqrt (+ (pow b 3) (pow c 3)))) (sqrt (- (* b b) (* c c))))
80.0ms
(* (exp (* (* a (sqrt (+ b c))) (sqrt (+ b c)))) (exp (* a d)))

prune735.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes307.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample1.0s

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