Average Error: 38.8 → 0.0
Time: 4.1s
Precision: 64
Internal Precision: 128
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[\left(x + 2\right) \cdot x\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 38.8

    \[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\left(2 + x\right) \cdot x}\]

  3. Final simplification0.0

    \[\leadsto \left(x + 2\right) \cdot x\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1) (+ x 1)) 1))

Details

Time bar (total: 4.0s)Debug log

sample97.0ms

Algorithm
intervals
Results
361×(pre true 80)
194×(body real 80)
72×(body real 1280)
50×(body real 640)
31×(body real 320)
14×(body real 160)

simplify452.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
451.0ms
(- (* (+ x 1) (+ x 1)) 1)

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 1 expressions with local error:

0.0b
(* (+ 2 x) x)

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
3×associate-*l*
2×add-cube-cbrt
2×associate-*l/
2×*-un-lft-identity
2×add-sqr-sqrt
1×add-log-exp
1×log1p-expm1-u
1×flip-+
1×*-commutative
1×add-exp-log
1×add-cbrt-cube
1×flip3-+
1×pow1
1×expm1-log1p-u
Counts
1 → 15
Calls
1 calls:
Slowest
5.0ms
(* (+ 2 x) x)

series30.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
30.0ms
(* (+ 2 x) x)

simplify258.0ms

Counts
7 → 18
Calls
7 calls:
Slowest
219.0ms
(* (- (* 2 2) (* x x)) x)
15.0ms
(* (+ (pow 2 3) (pow x 3)) x)
7.0ms
(+ (* 2 x) (pow x 2))
6.0ms
(+ (* 2 x) (pow x 2))
6.0ms
(+ (* 2 x) (pow x 2))

prune75.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 2 expressions with local error:

7.5b
(/ (* (- (* 2 2) (* x x)) x) (- 2 x))
0.1b
(* (- (* 2 2) (* x x)) x)

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-exp-log
4×associate-*l/
4×associate-*l*
4×add-cbrt-cube
3×flip--
3×flip3--
3×times-frac
3×associate-/r*
2×associate-/l/
2×add-log-exp
2×log1p-expm1-u
2×associate-/r/
2×pow1
2×expm1-log1p-u
1×difference-of-squares
1×associate-/l*
1×div-inv
1×*-commutative
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
2 → 41
Calls
2 calls:
Slowest
7.0ms
(/ (* (- (* 2 2) (* x x)) x) (- 2 x))
4.0ms
(* (- (* 2 2) (* x x)) x)

series72.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
41.0ms
(/ (* (- (* 2 2) (* x x)) x) (- 2 x))
32.0ms
(* (- (* 2 2) (* x x)) x)

simplify1.7s

Counts
27 → 47
Calls
27 calls:
Slowest
325.0ms
(* (- (pow (* 2 2) 3) (pow (* x x) 3)) x)
197.0ms
(/ (- (* 2 2) (* x x)) (* (cbrt (- 2 x)) (cbrt (- 2 x))))
183.0ms
(* (- (* (* 2 2) (* 2 2)) (* (* x x) (* x x))) x)
181.0ms
(* (- (* (* 2 2) (* 2 2)) (* (* x x) (* x x))) x)
165.0ms
(- (log (* (- (* 2 2) (* x x)) x)) (log (- 2 x)))

prune208.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes19.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample1.1s

Algorithm
intervals
Results
10762×(pre true 80)
5589×(body real 80)
2263×(body real 1280)
1689×(body real 640)
800×(body real 320)
421×(body real 160)