Average Error: 0.0 → 0.0
Time: 10.8s
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 2019022 +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: 10.3s)Debug log

sample37.0ms

Algorithm
intervals
Results
13.0ms256×pre80true
12.0ms256×body80valid

simplify46.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
46.0ms
(* a (+ (+ b c) d))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 1 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
3×associate-*r*
2×add-cube-cbrt
2×associate-*r/
2×*-un-lft-identity
2×add-sqr-sqrt
1×add-log-exp
1×log1p-expm1-u
1×flip-+
1×*-commutative
1×add-exp-log
1×add-cbrt-cube
1×flip3-+
1×distribute-lft-in
1×pow1
1×expm1-log1p-u
1×distribute-rgt-in
Counts
1 → 17
Calls
1 calls:
Slowest
7.0ms
(* a (+ (+ b c) d))

series25.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
25.0ms
(* a (+ (+ b c) d))

simplify274.0ms

Counts
10 → 20
Calls
10 calls:
Slowest
150.0ms
(+ (* (+ b c) (+ b c)) (- (* d d) (* (+ b c) d)))
46.0ms
(+ (* a d) (+ (* a b) (* a c)))
41.0ms
(+ (* a d) (+ (* a b) (* a c)))
11.0ms
(cbrt (+ (+ b c) d))
8.0ms
(- (+ b c) d)

prune172.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize14.0ms

Local error

Found 1 expressions with local error:

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

series30.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
30.0ms
(fma a (+ d b) (* c a))

simplify120.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
61.0ms
(+ (* a d) (+ (* a b) (* a c)))
58.0ms
(+ (* a d) (+ (* a b) (* a c)))
1.0ms
0

prune89.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize14.0ms

Local error

Found 4 expressions with local error:

5.9b
(/ (* a (- (* (+ b c) (+ b c)) (* d d))) (- (+ b c) d))
0.5b
(* a (- (* (+ b c) (+ b c)) (* d d)))
0.0b
(* (+ b c) (+ b c))
0.0b
(- (* (+ b c) (+ b c)) (* d d))

rewrite70.0ms

Algorithm
rewrite-expression-head
Rules
9×add-cube-cbrt
9×distribute-lft-in
9×*-un-lft-identity
9×pow1
9×add-sqr-sqrt
9×distribute-rgt-in
8×add-exp-log
8×add-cbrt-cube
7×associate-*r*
6×add-log-exp
6×flip-+
6×associate-*r/
6×associate--l+
6×flip3-+
4×flip--
4×log1p-expm1-u
4×flip3--
4×frac-times
4×expm1-log1p-u
3×prod-diff
3×associate-*l*
3×times-frac
3×sub-neg
3×associate-/r*
2×associate-/l/
2×difference-of-squares
2×*-commutative
2×associate-/r/
2×associate-*l/
1×associate-/l*
1×div-inv
1×cbrt-unprod
1×pow-prod-up
1×prod-exp
1×pow-prod-down
1×div-exp
1×diff-log
1×fma-neg
1×pow-plus
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×pow2
Counts
4 → 102
Calls
4 calls:
Slowest
26.0ms
(/ (* a (- (* (+ b c) (+ b c)) (* d d))) (- (+ b c) d))
26.0ms
(* a (- (* (+ b c) (+ b c)) (* d d)))
10.0ms
(- (* (+ b c) (+ b c)) (* d d))
6.0ms
(* (+ b c) (+ b c))

series288.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
124.0ms
(/ (* a (- (* (+ b c) (+ b c)) (* d d))) (- (+ b c) d))
101.0ms
(* a (- (* (+ b c) (+ b c)) (* d d)))
49.0ms
(- (* (+ b c) (+ b c)) (* d d))
14.0ms
(* (+ b c) (+ b c))

simplify6.5s

Counts
76 → 114
Calls
76 calls:
Slowest
634.0ms
(/ (- (* (+ b c) (+ b c)) (* d d)) (sqrt (- (+ b c) d)))
477.0ms
(- (log (* a (- (* (+ b c) (+ b c)) (* d d)))) (log (- (+ b c) d)))
317.0ms
(* (- (* b b) (* c c)) (+ (pow b 3) (pow c 3)))
287.0ms
(* (+ (* b b) (- (* c c) (* b c))) (+ (* b b) (- (* c c) (* b c))))
249.0ms
(/ (* (* (* a (- (* (+ b c) (+ b c)) (* d d))) (* a (- (* (+ b c) (+ b c)) (* d d)))) (* a (- (* (+ b c) (+ b c)) (* d d)))) (* (* (- (+ b c) d) (- (+ b c) d)) (- (+ b c) d)))

prune1.1s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes328.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample1.2s

Algorithm
intervals
Results
508.0ms8000×body80valid
480.0ms8000×pre80true