Average Error: 3.7 → 0.6
Time: 17.8s
Precision: 64
Internal Precision: 128
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[2 \cdot \frac{\left(\left(c + a\right) + \left(b - d\right)\right) \cdot \left(\left(c + b\right) + \left(d + a\right)\right)}{\sqrt[3]{\left(\left(\left(\left(c + b\right) + a\right) - d\right) \cdot \left(\left(\left(c + b\right) + a\right) - d\right)\right) \cdot \left(\left(\left(c + b\right) + a\right) - d\right)}}\]

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

Original3.7
Target3.9
Herbie0.6
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Derivation

  1. Initial program 3.7

    \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]

  2. Initial simplification3.2

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

  4. Applied associate-+r+3.1

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

  6. Applied associate-+l+2.8

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

  8. Applied add-cbrt-cube2.9

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

  10. Applied flip-+3.2

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

  11. Applied flip-+3.2

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

  12. Applied flip-+3.4

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

  13. Applied frac-times3.4

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

  14. Applied frac-times3.4

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

  15. Applied cbrt-div3.4

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

  16. Simplified0.6

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

  17. Final simplification0.6

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

Reproduce

herbie shell --seed 2018360 
(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: 13.9s)Debug log

start65.0ms

Algorithm
intervals

setup187.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 2.8b

localize28.0ms

Local error

Found 1 expressions with local error:

3.0b
(+ (+ a c) (+ b d))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
1 → 18
Calls

1 calls. Slowest were:

6.0ms
(+ (+ a c) (+ b d))

series26.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

26.0ms
(+ (+ a c) (+ b d))

simplify2.3s

Counts
13 → 21
Calls

13 calls. Slowest were:

499.0ms
(+ (* (+ (pow a 3) (pow c 3)) (- b d)) (* (+ (* a a) (- (* c c) (* a c))) (- (* b b) (* d d))))
419.0ms
(+ (* (- (* a a) (* c c)) (- b d)) (* (- a c) (- (* b b) (* d d))))
380.0ms
(* (+ (* a a) (- (* c c) (* a c))) (+ (* b b) (- (* d d) (* b d))))

prune305.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 1.9b

localize11.0ms

Local error

Found 2 expressions with local error:

2.7b
(+ (+ (+ a c) b) d)
0.3b
(+ (+ a c) b)

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
2 → 22
Calls

2 calls. Slowest were:

12.0ms
(+ (+ (+ a c) b) d)
2.0ms
(+ (+ a c) b)

series34.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

23.0ms
(+ (+ (+ a c) b) d)
11.0ms
(+ (+ a c) b)

simplify53.0ms

Counts
7 → 28
Calls

7 calls. Slowest were:

13.0ms
(+ b (+ c d))
10.0ms
(+ a (+ b c))
9.0ms
(+ b (+ c d))

prune360.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 1.7b

localize13.0ms

Local error

Found 1 expressions with local error:

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

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
1 → 11
Calls

1 calls. Slowest were:

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

series22.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

22.0ms
(+ (+ a (+ c b)) d)

simplify21.0ms

Counts
3 → 14
Calls

3 calls. Slowest were:

7.0ms
(+ a (+ b c))
7.0ms
(+ b (+ c d))
6.0ms
(+ b (+ c d))

prune180.0ms

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 1.6b

localize27.0ms

Local error

Found 4 expressions with local error:

2.7b
(+ (+ a (+ c b)) d)
2.7b
(+ (+ a (+ c b)) d)
2.7b
(+ (+ a (+ c b)) d)
0.6b
(cbrt (* (* (+ (+ a (+ c b)) d) (+ (+ a (+ c b)) d)) (+ (+ a (+ c b)) d)))

rewrite59.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

37.0ms
(cbrt (* (* (+ (+ a (+ c b)) d) (+ (+ a (+ c b)) d)) (+ (+ a (+ c b)) d)))
7.0ms
(+ (+ a (+ c b)) d)
7.0ms
(+ (+ a (+ c b)) d)

series188.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

78.0ms
(cbrt (* (* (+ (+ a (+ c b)) d) (+ (+ a (+ c b)) d)) (+ (+ a (+ c b)) d)))
38.0ms
(+ (+ a (+ c b)) d)
36.0ms
(+ (+ a (+ c b)) d)
35.0ms
(+ (+ a (+ c b)) d)

simplify8.3s

Counts
40 → 81
Calls

40 calls. Slowest were:

1.1s
(cbrt (* (+ (+ a (+ c b)) d) (+ (+ a (+ c b)) d)))
443.0ms
(cbrt (* (* (- (* (+ a (+ c b)) (+ a (+ c b))) (* d d)) (+ (+ a (+ c b)) d)) (+ (pow (+ a (+ c b)) 3) (pow d 3))))
437.0ms
(cbrt (* (* (+ (+ a (+ c b)) d) (+ (pow (+ a (+ c b)) 3) (pow d 3))) (+ (pow (+ a (+ c b)) 3) (pow d 3))))

prune1.1s

Pruning

14 alts after pruning (13 fresh and 1 done)

Merged error: 0.3b

regimes615.0ms

Accuracy

0% (0.2b remaining)

Error of 0.6b against oracle of 0.4b and baseline of 0.6b

bsearch5.0ms