Average Error: 3.6 → 0
Time: 9.6s
Precision: 64
Internal Precision: 128
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[\left(b + \left(c + \left(a + d\right)\right)\right) \cdot 2\]

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
Herbie0
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Derivation

  1. Initial program 3.6

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

  2. Initial simplification3.1

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

  4. Applied +-commutative3.1

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

  6. Applied associate-+l+2.8

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

  8. Applied associate-+r+0

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

  9. Final simplification0

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

Runtime

Time bar (total: 9.6s)Debug log

start95.0ms

Algorithm
intervals

setup325.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 2.9b

localize15.0ms

Local error

Found 1 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
1 → 23
Calls

1 calls. Slowest were:

8.0ms
(+ (+ a c) (+ b d))

series34.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

34.0ms
(+ (+ a c) (+ b d))

simplify2.8s

Counts
15 → 26
Calls

15 calls. Slowest were:

998.0ms
(+ (* (- (* a a) (* c c)) (- b d)) (* (- a c) (- (* b b) (* d d))))
619.0ms
(+ (* (+ (pow a 3) (pow c 3)) (- b d)) (* (+ (* a a) (- (* c c) (* a c))) (- (* b b) (* d d))))
474.0ms
(* (+ (* a a) (- (* c c) (* a c))) (+ (* b b) (- (* d d) (* b d))))

prune406.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 1.8b

localize11.0ms

Local error

Found 1 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
1 → 23
Calls

1 calls. Slowest were:

8.0ms
(+ (+ b d) (+ a c))

series33.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

33.0ms
(+ (+ b d) (+ a c))

simplify2.5s

Counts
15 → 26
Calls

15 calls. Slowest were:

570.0ms
(* (+ (* b b) (- (* d d) (* b d))) (+ (* a a) (- (* c c) (* a c))))
511.0ms
(+ (* (- (* b b) (* d d)) (- a c)) (* (- b d) (- (* a a) (* c c))))
502.0ms
(+ (* (+ (pow b 3) (pow d 3)) (- a c)) (* (+ (* b b) (- (* d d) (* b d))) (- (* a a) (* c c))))

prune437.0ms

Pruning

17 alts after pruning (16 fresh and 1 done)

Merged error: 0.8b

localize8.0ms

Local error

Found 1 expressions with local error:

0.9b
(+ d (+ a c))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 13
Calls

1 calls. Slowest were:

1.0ms
(+ d (+ a c))

series11.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

11.0ms
(+ d (+ a c))

simplify26.0ms

Counts
3 → 16
Calls

3 calls. Slowest were:

10.0ms
(+ a (+ c d))
9.0ms
(+ a (+ c d))
6.0ms
(+ a (+ c d))

prune163.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize11.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
0 → 0
Calls

0 calls. Slowest were:

series0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

simplify0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

prune7.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018359 +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))