Average Error: 0.0 → 0.0
Time: 22.1s
Precision: 64
Internal Precision: 128
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[a \cdot \left(b + a\right) + \left(b + a\right) \cdot b\]

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.0
Target0.0
Herbie0.0
\[\left(\left(b \cdot a + b \cdot b\right) + b \cdot a\right) + a \cdot a\]

Derivation

  1. Initial program 0.0

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

  3. Applied distribute-lft-in0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :pre (and (<= 5 a 10) (<= 0 b 0.001))

  :herbie-target
  (+ (+ (+ (* b a) (* b b)) (* b a)) (* a a))

  (* (+ a b) (+ a b)))

Details

Time bar (total: 21.7s)Debug log

sample34.0ms

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

simplify32.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
32.0ms
(* (+ a b) (+ a b))

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 1 expressions with local error:

0.0b
(* (+ a b) (+ a b))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
6×flip-+
6×flip3-+
6×pow1
4×frac-times
3×add-cube-cbrt
3×add-exp-log
3×associate-*l*
3×associate-*r*
3×add-cbrt-cube
3×*-un-lft-identity
3×add-sqr-sqrt
2×associate-*r/
2×associate-*l/
1×add-log-exp
1×log1p-expm1-u
1×cbrt-unprod
1×pow-prod-up
1×*-commutative
1×prod-exp
1×pow-prod-down
1×pow-plus
1×distribute-lft-in
1×expm1-log1p-u
1×distribute-rgt-in
1×pow2
Counts
1 → 32
Calls
1 calls:
Slowest
4.0ms
(* (+ a b) (+ a b))

series9.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
9.0ms
(* (+ a b) (+ a b))

simplify2.2s

Counts
28 → 35
Calls
28 calls:
Slowest
412.0ms
(* (- (* a a) (* b b)) (+ (pow a 3) (pow b 3)))
355.0ms
(* (+ (* a a) (- (* b b) (* a b))) (+ (* a a) (- (* b b) (* a b))))
243.0ms
(* (+ (pow a 3) (pow b 3)) (+ (pow a 3) (pow b 3)))
184.0ms
(* (- a b) (+ (* a a) (- (* b b) (* a b))))
173.0ms
(* (+ (pow a 3) (pow b 3)) (+ a b))

prune294.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize14.0ms

Local error

Found 3 expressions with local error:

0.0b
(* (+ a b) b)
0.0b
(* (+ a b) a)
0.0b
(+ (* (+ a b) a) (* (+ a b) b))

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
12×associate-*l/
7×flip-+
7×flip3-+
6×associate-*l*
5×add-log-exp
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×frac-add
3×log1p-expm1-u
3×add-exp-log
3×add-cbrt-cube
3×pow1
3×expm1-log1p-u
2×*-commutative
1×distribute-lft-out
1×fma-def
1×sum-log
1×+-commutative
Counts
3 → 49
Calls
3 calls:
Slowest
9.0ms
(+ (* (+ a b) a) (* (+ a b) b))
5.0ms
(* (+ a b) b)
4.0ms
(* (+ a b) a)

series70.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
24.0ms
(* (+ a b) b)
23.0ms
(+ (* (+ a b) a) (* (+ a b) b))
22.0ms
(* (+ a b) a)

simplify2.8s

Counts
29 → 58
Calls
29 calls:
Slowest
717.0ms
(* (- (* a a) (* b b)) a)
422.0ms
(* (- (* a a) (* b b)) b)
303.0ms
(* (+ (* a a) (- (* b b) (* a b))) (+ (* a a) (- (* b b) (* a b))))
246.0ms
(* (- a b) (+ (* a a) (- (* b b) (* a b))))
165.0ms
(+ (* (* (+ (pow a 3) (pow b 3)) a) (+ (* a a) (- (* b b) (* a b)))) (* (+ (* a a) (- (* b b) (* a b))) (* (+ (pow a 3) (pow b 3)) b)))

prune428.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize65.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (* (+ b a) (- a b)) (fma b (- b a) (* a a)))
0.4b
(/ (fma (* (* (+ b a) (- a b)) (fma b (- b a) (* a a))) b (* (* (- a b) a) (fma b (* b b) (pow a 3)))) (* (- a b) (fma b (- b a) (* a a))))
0.4b
(* (* (- a b) a) (fma b (* b b) (pow a 3)))
0.3b
(* (- a b) (fma b (- b a) (* a a)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
18×add-exp-log
18×associate-*l/
18×add-cbrt-cube
13×pow1
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
9×associate-*r*
6×flip--
6×fma-udef
6×cbrt-unprod
6×prod-exp
6×flip3--
5×pow-prod-down
5×associate-*l*
4×add-log-exp
4×log1p-expm1-u
4×frac-times
4×expm1-log1p-u
3×associate-/l*
3×flip-+
3×*-commutative
3×times-frac
3×flip3-+
3×distribute-lft-in
3×distribute-rgt-in
2×associate-/r/
2×associate-*r/
2×div-exp
2×cbrt-undiv
1×div-inv
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
4 → 99
Calls
4 calls:
Slowest
11.0ms
(* (* (+ b a) (- a b)) (fma b (- b a) (* a a)))
8.0ms
(* (* (- a b) a) (fma b (* b b) (pow a 3)))
6.0ms
(/ (fma (* (* (+ b a) (- a b)) (fma b (- b a) (* a a))) b (* (* (- a b) a) (fma b (* b b) (pow a 3)))) (* (- a b) (fma b (- b a) (* a a))))
4.0ms
(* (- a b) (fma b (- b a) (* a a)))

series222.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
95.0ms
(/ (fma (* (* (+ b a) (- a b)) (fma b (- b a) (* a a))) b (* (* (- a b) a) (fma b (* b b) (pow a 3)))) (* (- a b) (fma b (- b a) (* a a))))
60.0ms
(* (* (+ b a) (- a b)) (fma b (- b a) (* a a)))
40.0ms
(* (* (- a b) a) (fma b (* b b) (pow a 3)))
26.0ms
(* (- a b) (fma b (- b a) (* a a)))

simplify12.7s

Counts
70 → 111
Calls
70 calls:
Slowest
637.0ms
(* (* (+ (pow b 3) (pow a 3)) (- (* a a) (* b b))) (fma b (- b a) (* a a)))
601.0ms
(* (* (- (* b b) (* a a)) (- a b)) (fma b (- b a) (* a a)))
586.0ms
(/ (fma (* (* (+ b a) (- a b)) (fma b (- b a) (* a a))) b (* (* (- a b) a) (fma b (* b b) (pow a 3)))) (fma b (- b a) (* a a)))
557.0ms
(sqrt (fma (* (* (+ b a) (- a b)) (fma b (- b a) (* a a))) b (* (* (- a b) a) (fma b (* b b) (pow a 3)))))
508.0ms
(/ (cbrt (fma (* (* (+ b a) (- a b)) (fma b (- b a) (* a a))) b (* (* (- a b) a) (fma b (* b b) (pow a 3))))) (fma b (- b a) (* a a)))

prune1.6s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes65.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample1.1s

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