Average Error: 0.5 → 0.4
Time: 6.3s
Precision: 64
Internal Precision: 128
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\frac{\frac{-1}{8}}{x} - \left(\frac{1}{2} - x\right)\]

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. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{x - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{-1}{8}}{x} - \left(\frac{1}{2} - x\right)}\]

  4. Final simplification0.4

    \[\leadsto \frac{\frac{-1}{8}}{x} - \left(\frac{1}{2} - x\right)\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))

Details

Time bar (total: 6.1s)Debug log

sample251.0ms

Algorithm
intervals

simplify38.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

37.0ms
(* (sqrt (- x 1)) (sqrt x))

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize24.0ms

Local error

Found 1 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
1 → 26
Calls

1 calls. Slowest were:

12.0ms
(* (sqrt (- x 1)) (sqrt x))

series54.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

54.0ms
(* (sqrt (- x 1)) (sqrt x))

simplify676.0ms

Counts
18 → 29
Calls

18 calls. Slowest were:

428.0ms
(* (* (* (sqrt (- x 1)) (sqrt (- x 1))) (sqrt (- x 1))) (* (* (sqrt x) (sqrt x)) (sqrt x)))
113.0ms
(- x (+ (* 1/8 (/ 1 x)) 1/2))
44.0ms
(* (sqrt (- (* x x) (* 1 1))) (sqrt x))

prune228.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize17.0ms

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:

prune4.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize10.0ms

Local error

Found 2 expressions with local error:

30.7b
(sqrt (* (- x 1) x))
0.0b
(* (- x 1) x)

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 28
Calls

2 calls. Slowest were:

5.0ms
(* (- x 1) x)
3.0ms
(sqrt (* (- x 1) x))

series53.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

30.0ms
(* (- x 1) x)
23.0ms
(sqrt (* (- x 1) x))

simplify557.0ms

Counts
13 → 34
Calls

13 calls. Slowest were:

156.0ms
(sqrt (* (- (* x x) (* 1 1)) x))
123.0ms
(* (- (* x x) (* 1 1)) x)
115.0ms
(- x (+ (* 1/8 (/ 1 x)) 1/2))

prune268.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes35.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch1.0ms

end0.0ms

sample3.8s

Algorithm
intervals