Average Error: 0.0 → 0.0
Time: 5.2s
Precision: 64
Internal Precision: 128
\[a \cdot a - b \cdot b\]
↓
\[\left(b + a\right) \cdot \left(a - b\right)\]

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

Derivation

  1. Initial program 0.0

    \[a \cdot a - b \cdot b\]
  2. Using strategy rm

  3. Applied difference-of-squares0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (a b)
  :name "Difference of squares"

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

  (- (* a a) (* b b)))

Details

Time bar (total: 4.8s)Debug log

sample30.0ms

Algorithm
intervals

simplify12.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
12.0ms
(- (* a a) (* b b))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 1 expressions with local error:

0.0b
(- (* a a) (* b b))

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
3×add-log-exp
1×difference-of-squares
1×flip--
1×log1p-expm1-u
1×add-cube-cbrt
1×add-exp-log
1×prod-diff
1×diff-log
1×fma-neg
1×add-cbrt-cube
1×flip3--
1×*-un-lft-identity
1×sub-neg
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 17
Calls
1 calls:
Slowest
2.0ms
(- (* a a) (* b b))

series24.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
24.0ms
(- (* a a) (* b b))

simplify71.0ms

Counts
8 → 20
Calls
8 calls:
Slowest
21.0ms
(/ (exp (* a a)) (exp (* b b)))
16.0ms
(fma a a (- (* b b)))
11.0ms
(- (pow a 2) (pow b 2))
11.0ms
(- (pow a 2) (pow b 2))
5.0ms
(- (pow a 2) (pow b 2))

prune129.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize11.0ms

Local error

Found 1 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
4×frac-times
3×flip--
3×flip-+
3×add-cube-cbrt
3×add-exp-log
3×associate-*l*
3×associate-*r*
3×add-cbrt-cube
3×flip3--
3×flip3-+
3×*-un-lft-identity
3×pow1
3×add-sqr-sqrt
2×associate-*r/
2×associate-*l/
2×distribute-lft-in
2×sub-neg
2×distribute-rgt-in
1×add-log-exp
1×log1p-expm1-u
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×expm1-log1p-u
Counts
1 → 31
Calls
1 calls:
Slowest
6.0ms
(* (+ a b) (- a b))

series23.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
23.0ms
(* (+ a b) (- a b))

simplify2.3s

Counts
27 → 34
Calls
27 calls:
Slowest
477.0ms
(* (+ (* a a) (- (* b b) (* a b))) (+ (* a a) (+ (* b b) (* a b))))
371.0ms
(* (+ (pow a 3) (pow b 3)) (- (pow a 3) (pow b 3)))
217.0ms
(* (- a b) (+ (* a a) (+ (* b b) (* a b))))
192.0ms
(* (+ (pow a 3) (pow b 3)) (- a b))
166.0ms
(* (- (* a a) (* b b)) (- (pow a 3) (pow b 3)))

prune331.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize12.0ms

Local error

Found 1 expressions with local error:

0.0b
(fma a a (- (* b b)))

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×log1p-expm1-u
1×fma-udef
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
0.0ms
(fma a a (- (* b b)))

series34.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
34.0ms
(fma a a (- (* b b)))

simplify37.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
14.0ms
(- (pow a 2) (pow b 2))
11.0ms
(- (pow a 2) (pow b 2))
11.0ms
(- (pow a 2) (pow b 2))

prune104.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes85.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0b and baseline of 0.0b

bsearch1.0ms

end0.0ms

sample1.6s

Algorithm
intervals