Average Error: 0.0 → 0.0
Time: 4.2s
Precision: 64
Internal Precision: 128
\[a \cdot \left(\left(b + c\right) + d\right)\]
↓
\[(\left(c + b\right) \cdot a + \left(a \cdot d\right))_*\]

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. Using strategy rm

  3. Applied distribute-rgt-in0.0

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

  5. Applied fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019016 +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: 3.7s)Debug log

sample30.0ms

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

simplify53.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
53.0ms
(* 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)

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×associate-*r*
3×fma-def
2×add-log-exp
2×log1p-expm1-u
2×flip-+
2×add-exp-log
2×associate-*r/
2×add-cbrt-cube
2×flip3-+
2×pow1
2×expm1-log1p-u
1×*-commutative
1×distribute-lft-in
1×associate-+l+
1×+-commutative
1×distribute-rgt-in
Counts
2 → 33
Calls
2 calls:
Slowest
4.0ms
(* a (+ (+ b c) d))
3.0ms
(+ (+ b c) d)

series37.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
20.0ms
(* a (+ (+ b c) d))
17.0ms
(+ (+ b c) d)

simplify349.0ms

Counts
15 → 39
Calls
15 calls:
Slowest
165.0ms
(+ (* (+ b c) (+ b c)) (- (* d d) (* (+ b c) d)))
47.0ms
(+ (* a d) (+ (* a b) (* a c)))
43.0ms
(+ (* a d) (+ (* a b) (* a c)))
16.0ms
(- (+ b c) d)
13.0ms
(+ b (+ c d))

prune392.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 1 expressions with local error:

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

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×log1p-expm1-u
1×fma-udef
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
0.0ms
(fma a (+ d b) (* c a))

series34.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
34.0ms
(fma a (+ d b) (* c a))

simplify104.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
59.0ms
(+ (* a d) (+ (* a b) (* a c)))
44.0ms
(+ (* a d) (+ (* a b) (* a c)))
0.0ms
0

prune113.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 2 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
3×add-cube-cbrt
3×associate-*l*
3×*-un-lft-identity
3×add-sqr-sqrt
2×log1p-expm1-u
2×flip-+
2×add-exp-log
2×associate-*l/
2×add-cbrt-cube
2×flip3-+
2×pow1
2×expm1-log1p-u
1×*-commutative
1×distribute-rgt-out
1×fma-def
1×sum-log
1×+-commutative
Counts
2 → 30
Calls
2 calls:
Slowest
4.0ms
(+ (* (+ b c) a) (* d a))
3.0ms
(* (+ b c) a)

series39.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
24.0ms
(+ (* (+ b c) a) (* d a))
15.0ms
(* (+ b c) a)

simplify467.0ms

Counts
13 → 36
Calls
13 calls:
Slowest
200.0ms
(* (- (* b b) (* c c)) a)
114.0ms
(* (exp (* (+ b c) a)) (exp (* d a)))
44.0ms
(+ (* a d) (+ (* a b) (* a c)))
42.0ms
(+ (* a d) (+ (* a b) (* a c)))
29.0ms
(* (+ (pow b 3) (pow c 3)) a)

prune352.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize4.0ms

Local error

Found 1 expressions with local error:

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

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×log1p-expm1-u
1×fma-udef
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
0.0ms
(fma (+ b c) a (* d a))

series22.0ms

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

simplify107.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
63.0ms
(+ (* a d) (+ (* a b) (* a c)))
44.0ms
(+ (* a d) (+ (* a b) (* a c)))
0.0ms
0

prune159.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes238.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample1.1s

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