Average Error: 3.6 → 2.7
Time: 13.4s
Precision: 64
Internal Precision: 128
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[\sqrt[3]{\left(\left(d + \left(\left(b + c\right) + a\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot 2\]

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
Target3.8
Herbie2.7
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

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+2.7

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

  5. Applied add-cbrt-cube2.9

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)}} \cdot 2\]
  6. Using strategy rm

  7. Applied associate-+r+2.7

    \[\leadsto \sqrt[3]{\left(\color{blue}{\left(\left(a + \left(b + c\right)\right) + d\right)} \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot 2\]

  8. Final simplification2.7

    \[\leadsto \sqrt[3]{\left(\left(d + \left(\left(b + c\right) + a\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot 2\]

Reproduce

herbie shell --seed 2019022 +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: 11.0s)Debug log

sample32.0ms

Algorithm
intervals
Results
10.0ms256×body80valid
10.0ms256×pre80true

simplify204.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
204.0ms
(* (+ a (+ b (+ c d))) 2)

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 3.7b

localize26.0ms

Local error

Found 2 expressions with local error:

2.7b
(+ a (+ b (+ c d)))
0.5b
(+ 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))

series49.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
33.0ms
(+ a (+ b (+ c d)))
16.0ms
(+ b (+ c d))

simplify63.0ms

Counts
7 → 32
Calls
7 calls:
Slowest
16.0ms
(+ b (+ c d))
12.0ms
(+ b (+ c d))
11.0ms
(+ a (+ b c))
7.0ms
(+ b (+ c d))
6.0ms
(+ b (+ c d))

prune305.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 2.3b

localize18.0ms

Local error

Found 1 expressions with local error:

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

rewrite4.0ms

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
4.0ms
(+ a (+ (+ b c) d))

series22.0ms

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

simplify21.0ms

Counts
3 → 16
Calls
3 calls:
Slowest
8.0ms
(+ a (+ b c))
7.0ms
(+ b (+ c d))
6.0ms
(+ b (+ c d))

prune164.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 2.0b

localize7.0ms

Local error

Found 1 expressions with local error:

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

rewrite5.0ms

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
5.0ms
(+ (+ a (+ b c)) d)

series34.0ms

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

simplify35.0ms

Counts
5 → 19
Calls
5 calls:
Slowest
9.0ms
(* (cbrt (+ a (+ b c))) (cbrt (+ a (+ b c))))
7.0ms
(+ b (+ c d))
6.0ms
(sqrt (+ a (+ b c)))
6.0ms
(+ a (+ b c))
6.0ms
(+ b (+ c d))

prune209.0ms

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 1.7b

localize24.0ms

Local error

Found 4 expressions with local error:

2.7b
(+ a (+ (+ b c) d))
2.7b
(+ a (+ (+ b c) d))
2.7b
(+ a (+ (+ b c) d))
0.5b
(cbrt (* (* (+ a (+ (+ b c) d)) (+ a (+ (+ b c) d))) (+ a (+ (+ b c) d))))

rewrite49.0ms

Algorithm
rewrite-expression-head
Rules
30×flip-+
30×flip3-+
28×frac-times
26×cbrt-div
14×associate-*l/
8×associate-*r/
4×add-log-exp
4×log1p-expm1-u
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×expm1-log1p-u
4×add-sqr-sqrt
3×associate-+r+
3×+-commutative
1×rem-cbrt-cube
1×pow3
1×pow1/3
1×cbrt-prod
Counts
4 → 77
Calls
4 calls:
Slowest
31.0ms
(cbrt (* (* (+ a (+ (+ b c) d)) (+ a (+ (+ b c) d))) (+ a (+ (+ b c) d))))
5.0ms
(+ a (+ (+ b c) d))
5.0ms
(+ a (+ (+ b c) d))
5.0ms
(+ a (+ (+ b c) d))

series136.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
58.0ms
(cbrt (* (* (+ a (+ (+ b c) d)) (+ a (+ (+ b c) d))) (+ a (+ (+ b c) d))))
34.0ms
(+ a (+ (+ b c) d))
24.0ms
(+ a (+ (+ b c) d))
20.0ms
(+ a (+ (+ b c) d))

simplify6.4s

Counts
39 → 89
Calls
39 calls:
Slowest
376.0ms
(cbrt (* (* (- (* a a) (* (+ (+ b c) d) (+ (+ b c) d))) (+ a (+ (+ b c) d))) (+ (pow a 3) (pow (+ (+ b c) d) 3))))
367.0ms
(cbrt (* (* (+ a (+ (+ b c) d)) (+ (pow a 3) (pow (+ (+ b c) d) 3))) (+ (pow a 3) (pow (+ (+ b c) d) 3))))
344.0ms
(cbrt (* (* (- (* a a) (* (+ (+ b c) d) (+ (+ b c) d))) (+ (pow a 3) (pow (+ (+ b c) d) 3))) (+ (pow a 3) (pow (+ (+ b c) d) 3))))
327.0ms
(cbrt (* (* (+ a (+ (+ b c) d)) (+ (pow a 3) (pow (+ (+ b c) d) 3))) (+ a (+ (+ b c) d))))
321.0ms
(cbrt (* (* (- (* a a) (* (+ (+ b c) d) (+ (+ b c) d))) (+ (pow a 3) (pow (+ (+ b c) d) 3))) (+ a (+ (+ b c) d))))

prune896.0ms

Pruning

23 alts after pruning (22 fresh and 1 done)

Merged error: 1.5b

regimes927.0ms

Accuracy

0% (1.2b remaining)

Error of 2.7b against oracle of 1.5b and baseline of 2.7b

bsearch2.0ms

end0.0ms

sample1.4s

Algorithm
intervals
Results
743.0ms8000×pre80true
424.0ms8000×body80valid