Average Error: 0.5 → 0.5
Time: 7.2s
Precision: 64
Internal Precision: 128
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\sqrt{x - 1} \cdot \sqrt{x}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\sqrt{x - 1} \cdot \sqrt{x}\]

  2. Final simplification0.5

    \[\leadsto \sqrt{x - 1} \cdot \sqrt{x}\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))

Details

Time bar (total: 6.9s)Debug log

sample344.0ms

Algorithm
intervals
Results
1040×(pre true 80)
783×(body nan 80)
256×(body real 80)
1×(body real 160)

simplify35.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
35.0ms
(* (sqrt (- x 1)) (sqrt x))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize15.0ms

Local error

Found 1 expressions with local error:

0.5b
(* (sqrt (- x 1)) (sqrt x))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
6×associate-*l*
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×sqrt-prod
3×add-exp-log
3×associate-*r*
3×add-cbrt-cube
3×pow1
2×associate-*l/
2×pow-prod-down
2×sqrt-div
2×pow1/2
1×add-log-exp
1×flip--
1×log1p-expm1-u
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×sqrt-unprod
1×flip3--
1×expm1-log1p-u
Counts
1 → 26
Calls
1 calls:
Slowest
7.0ms
(* (sqrt (- x 1)) (sqrt x))

series33.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
32.0ms
(* (sqrt (- x 1)) (sqrt x))

simplify1.4s

Counts
18 → 29
Calls
18 calls:
Slowest
1.1s
(* (* (* (sqrt (- x 1)) (sqrt (- x 1))) (sqrt (- x 1))) (* (* (sqrt x) (sqrt x)) (sqrt x)))
79.0ms
(* (sqrt (- (* x x) (* 1 1))) (sqrt x))
76.0ms
(- x (+ (* 1/8 (/ 1 x)) 1/2))
45.0ms
(* (- x 1) x)
21.0ms
(* (sqrt (- (pow x 3) (pow 1 3))) (sqrt x))

prune238.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize15.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
Counts
0 → 0
Calls
0 calls:
Slowest

series0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

simplify0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

prune4.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize9.0ms

Local error

Found 1 expressions with local error:

29.3b
(sqrt (* (- x 1) x))

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
2×associate-*l/
2×sqrt-div
1×add-log-exp
1×flip--
1×log1p-expm1-u
1×sqrt-prod
1×add-cube-cbrt
1×add-exp-log
1×pow1/2
1×add-cbrt-cube
1×flip3--
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 13
Calls
1 calls:
Slowest
3.0ms
(sqrt (* (- x 1) x))

series19.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
19.0ms
(sqrt (* (- x 1) x))

simplify296.0ms

Counts
6 → 16
Calls
6 calls:
Slowest
125.0ms
(sqrt (* (- (* x x) (* 1 1)) x))
89.0ms
(- x (+ (* 1/8 (/ 1 x)) 1/2))
59.0ms
(- (+ (* 1/8 (/ 1 x)) 1/2) x)
18.0ms
(sqrt (* (- (pow x 3) (pow 1 3)) x))
4.0ms
(sqrt (- x 1))

prune139.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes72.0ms

Accuracy

0% (0.5b remaining)

Error of 0.5b against oracle of 0.0b and baseline of 0.5b

bsearch1.0ms

end0.0ms

sample4.2s

Algorithm
intervals
Results
32215×(pre true 80)
24215×(body nan 80)
7992×(body real 80)
8×(body real 160)