Average Error: 3.6 → 2.7
Time: 24.3s
Precision: 64
Internal Precision: 128
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[\sqrt[3]{\left(\left(\left(\left(b + c\right) + d\right) + a\right) \cdot \left(\left(\left(b + c\right) + a\right) + d\right)\right) \cdot \left(\left(\left(b + c\right) + a\right) + d\right)} \cdot 2\]

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
Target3.9
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+2.8

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

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

  9. Applied associate-+r+2.7

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

  10. Final simplification2.7

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

Reproduce

herbie shell --seed 2019016 
(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: 20.7s)Debug log

sample63.0ms

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

simplify200.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
200.0ms
(* (+ a (+ b (+ c d))) 2)

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 3.6b

localize13.0ms

Local error

Found 2 expressions with local error:

2.7b
(+ a (+ b (+ c d)))
0.4b
(+ b (+ c d))

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
2×add-log-exp
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×add-sqr-sqrt
Counts
2 → 22
Calls
2 calls:
Slowest
1.0ms
(+ a (+ b (+ c d)))
1.0ms
(+ b (+ c d))

series51.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
33.0ms
(+ a (+ b (+ c d)))
18.0ms
(+ b (+ c d))

simplify78.0ms

Counts
7 → 28
Calls
7 calls:
Slowest
13.0ms
(+ b (+ c d))
12.0ms
(+ b (+ c d))
12.0ms
(+ b (+ c d))
12.0ms
(+ a (+ b c))
12.0ms
(+ b (+ c d))

prune319.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 2.2b

localize12.0ms

Local error

Found 1 expressions with local error:

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

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
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×add-sqr-sqrt
Counts
1 → 11
Calls
1 calls:
Slowest
3.0ms
(+ a (+ (+ b c) d))

series21.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
21.0ms
(+ a (+ (+ b c) d))

simplify19.0ms

Counts
3 → 14
Calls
3 calls:
Slowest
6.0ms
(+ a (+ b c))
6.0ms
(+ b (+ c d))
6.0ms
(+ b (+ c d))

prune165.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 2.0b

localize22.0ms

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))
0.6b
(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×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
3×associate-+r+
3×+-commutative
1×rem-cbrt-cube
1×pow3
1×pow1/3
1×cbrt-prod
Counts
4 → 69
Calls
4 calls:
Slowest
32.0ms
(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))

series184.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
76.0ms
(cbrt (* (* (+ a (+ (+ b c) d)) (+ a (+ (+ b c) d))) (+ a (+ (+ b c) d))))
39.0ms
(+ a (+ (+ b c) d))
36.0ms
(+ a (+ (+ b c) d))
34.0ms
(+ a (+ (+ b c) d))

simplify7.3s

Counts
39 → 81
Calls
39 calls:
Slowest
918.0ms
(cbrt (* (+ a (+ (+ b c) d)) (+ a (+ (+ b c) d))))
391.0ms
(cbrt (* (* (- (* a a) (* (+ (+ b c) d) (+ (+ b c) d))) (+ a (+ (+ b c) d))) (+ (pow a 3) (pow (+ (+ b c) d) 3))))
372.0ms
(cbrt (* (* (- (* a a) (* (+ (+ b c) d) (+ (+ b c) d))) (+ (pow a 3) (pow (+ (+ b c) d) 3))) (+ a (+ (+ b c) d))))
330.0ms
(cbrt (* (* (+ a (+ (+ b c) d)) (+ (pow a 3) (pow (+ (+ b c) d) 3))) (+ (pow a 3) (pow (+ (+ b c) d) 3))))
329.0ms
(cbrt (* (* (+ a (+ (+ b c) d)) (+ (pow a 3) (pow (+ (+ b c) d) 3))) (- (* a a) (* (+ (+ b c) d) (+ (+ b c) d)))))

prune1.1s

Pruning

22 alts after pruning (21 fresh and 1 done)

Merged error: 1.7b

localize13.0ms

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)
0.6b
(cbrt (* (* (+ a (+ (+ b c) d)) (+ a (+ (+ b c) d))) (+ (+ a (+ b c)) d)))

rewrite43.0ms

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×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
3×+-commutative
2×associate-+r+
1×pow1/3
1×associate-+l+
1×cbrt-prod
Counts
4 → 68
Calls
4 calls:
Slowest
28.0ms
(cbrt (* (* (+ a (+ (+ b c) d)) (+ a (+ (+ b c) d))) (+ (+ a (+ b c)) d)))
6.0ms
(+ a (+ (+ b c) d))
4.0ms
(+ (+ a (+ b c)) d)
3.0ms
(+ a (+ (+ b c) d))

series189.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
102.0ms
(cbrt (* (* (+ a (+ (+ b c) d)) (+ a (+ (+ b c) d))) (+ (+ a (+ b c)) d)))
35.0ms
(+ a (+ (+ b c) d))
31.0ms
(+ a (+ (+ b c) d))
20.0ms
(+ (+ a (+ b c)) d)

simplify7.5s

Counts
39 → 80
Calls
39 calls:
Slowest
958.0ms
(cbrt (* (+ a (+ (+ b c) d)) (+ a (+ (+ b c) d))))
596.0ms
(cbrt (* (* (+ a (+ (+ b c) d)) (- (* a a) (* (+ (+ b c) d) (+ (+ b c) d)))) (+ (pow (+ a (+ b c)) 3) (pow d 3))))
493.0ms
(cbrt (* (* (+ a (+ (+ b c) d)) (+ (pow a 3) (pow (+ (+ b c) d) 3))) (+ (pow (+ a (+ b c)) 3) (pow d 3))))
329.0ms
(cbrt (* (* (- (* a a) (* (+ (+ b c) d) (+ (+ b c) d))) (+ (pow a 3) (pow (+ (+ b c) d) 3))) (+ (pow (+ a (+ b c)) 3) (pow d 3))))
318.0ms
(cbrt (* (* (- (* a a) (* (+ (+ b c) d) (+ (+ b c) d))) (+ a (+ (+ b c) d))) (+ (pow (+ a (+ b c)) 3) (pow d 3))))

prune800.0ms

Pruning

28 alts after pruning (27 fresh and 1 done)

Merged error: 1.4b

regimes1.4s

Accuracy

0% (1.3b remaining)

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

bsearch4.0ms

end0.0ms

sample1.1s

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