Average Error: 0.4 → 0.2
Time: 11.4s
Precision: 64
Internal Precision: 128
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[e + \left(c + \left(\left(b + a\right) + d\right)\right)\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.4
Target0.2
Herbie0.2
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e\]

Derivation

  1. Initial program 0.4

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

  3. Applied associate-+l+0.3

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

  4. Taylor expanded around 0 0.3

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

  6. Applied associate-+l+0.2

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

  8. Applied associate-+l+0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019016 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :pre (<= 1 a 2 b 4 c 8 d 16 e 32)

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

  (+ (+ (+ (+ e d) c) b) a))

Details

Time bar (total: 10.5s)Debug log

sample31.0ms

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

simplify261.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
261.0ms
(+ (+ (+ (+ e d) c) b) a)

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize20.0ms

Local error

Found 3 expressions with local error:

0.2b
(+ (+ (+ (+ e d) c) b) a)
0.2b
(+ (+ (+ e d) c) b)
0.1b
(+ (+ e d) c)

rewrite68.0ms

Algorithm
rewrite-expression-head
Rules
3×add-log-exp
3×flip-+
3×add-cube-cbrt
3×add-exp-log
3×add-cbrt-cube
3×flip3-+
3×*-un-lft-identity
3×associate-+l+
3×+-commutative
3×pow1
3×add-sqr-sqrt
Counts
3 → 33
Calls
3 calls:
Slowest
53.0ms
(+ (+ (+ (+ e d) c) b) a)
11.0ms
(+ (+ (+ e d) c) b)
3.0ms
(+ (+ e d) c)

series67.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
37.0ms
(+ (+ (+ (+ e d) c) b) a)
20.0ms
(+ (+ (+ e d) c) b)
10.0ms
(+ (+ e d) c)

simplify67.0ms

Counts
11 → 42
Calls
11 calls:
Slowest
7.0ms
(+ e (+ c d))
7.0ms
(+ b (+ c d))
7.0ms
(+ e (+ c d))
6.0ms
(+ e (+ c d))
6.0ms
(+ e (+ c d))

prune500.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 2 expressions with local error:

0.2b
(+ (+ (+ e d) c) (+ b a))
0.1b
(+ (+ e d) c)

rewrite27.0ms

Algorithm
rewrite-expression-head
Rules
6×flip-+
6×flip3-+
4×add-log-exp
4×frac-add
4×*-un-lft-identity
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×associate-+l+
2×+-commutative
2×pow1
2×add-sqr-sqrt
1×distribute-lft-out
1×sum-log
1×associate-+r+
Counts
2 → 29
Calls
2 calls:
Slowest
22.0ms
(+ (+ (+ e d) c) (+ b a))
4.0ms
(+ (+ e d) c)

series50.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
40.0ms
(+ (+ (+ e d) c) (+ b a))
10.0ms
(+ (+ e d) c)

simplify2.0s

Counts
17 → 35
Calls
17 calls:
Slowest
407.0ms
(* (- (+ e d) c) (+ (* b b) (- (* a a) (* b a))))
312.0ms
(* (- (+ e d) c) (- b a))
250.0ms
(* (+ (* (+ e d) (+ e d)) (- (* c c) (* (+ e d) c))) (- b a))
211.0ms
(* (exp (+ (+ e d) c)) (exp (+ b a)))
190.0ms
(+ (+ (+ e d) c) (+ b a))

prune429.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize8.0ms

Local error

Found 2 expressions with local error:

0.2b
(+ (+ e (+ c d)) (+ b a))
0.2b
(+ e (+ c d))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
6×flip-+
6×flip3-+
4×add-log-exp
4×frac-add
4×*-un-lft-identity
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×associate-+r+
2×+-commutative
2×pow1
2×add-sqr-sqrt
1×distribute-lft-out
1×sum-log
1×associate-+l+
Counts
2 → 29
Calls
2 calls:
Slowest
9.0ms
(+ (+ e (+ c d)) (+ b a))
1.0ms
(+ e (+ c d))

series87.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
69.0ms
(+ (+ e (+ c d)) (+ b a))
18.0ms
(+ e (+ c d))

simplify1.8s

Counts
16 → 35
Calls
16 calls:
Slowest
273.0ms
(* (+ (* e e) (- (* (+ c d) (+ c d)) (* e (+ c d)))) (- b a))
258.0ms
(* (- e (+ c d)) (- b a))
244.0ms
(* (exp (+ e (+ c d))) (exp (+ b a)))
226.0ms
(* (- e (+ c d)) (+ (* b b) (- (* a a) (* b a))))
183.0ms
(+ (* (+ (pow e 3) (pow (+ c d) 3)) (+ (* b b) (- (* a a) (* b a)))) (* (+ (* e e) (- (* (+ c d) (+ c d)) (* e (+ c d)))) (+ (pow b 3) (pow a 3))))

prune380.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

localize10.0ms

Local error

Found 2 expressions with local error:

0.3b
(+ (+ c d) (+ b a))
0.2b
(+ e (+ (+ c d) (+ b a)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
6×flip-+
6×flip3-+
4×add-log-exp
4×frac-add
4×*-un-lft-identity
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×associate-+r+
2×+-commutative
2×pow1
2×add-sqr-sqrt
1×distribute-lft-out
1×sum-log
1×associate-+l+
Counts
2 → 29
Calls
2 calls:
Slowest
7.0ms
(+ (+ c d) (+ b a))
6.0ms
(+ e (+ (+ c d) (+ b a)))

series80.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
48.0ms
(+ e (+ (+ c d) (+ b a)))
32.0ms
(+ (+ c d) (+ b a))

simplify1.9s

Counts
17 → 35
Calls
17 calls:
Slowest
418.0ms
(+ (* (+ (pow c 3) (pow d 3)) (- b a)) (* (+ (* c c) (- (* d d) (* c d))) (- (* b b) (* a a))))
365.0ms
(+ (* (- (* c c) (* d d)) (- b a)) (* (- c d) (- (* b b) (* a a))))
319.0ms
(* (+ (* c c) (- (* d d) (* c d))) (+ (* b b) (- (* a a) (* b a))))
305.0ms
(* (+ (* c c) (- (* d d) (* c d))) (- b a))
162.0ms
(* (- c d) (+ (* b b) (- (* a a) (* b a))))

prune353.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

regimes560.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch1.0ms

end0.0ms

sample1.7s

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