Average Error: 0.5 → 0.4
Time: 7.6s
Precision: 64
Internal Precision: 128
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\frac{\frac{-1}{8}}{x} + \left(x + \frac{-1}{2}\right)\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  2. Initial simplification0.5

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

  3. Taylor expanded around inf 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2018360 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))

Details

Time bar (total: 2.2s)Debug log

start316.0ms

Algorithm
intervals

setup40.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize12.0ms

Local error

Found 1 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
1 → 24
Calls

1 calls. Slowest were:

6.0ms
(* (sqrt (+ x -1)) (sqrt x))

series66.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

66.0ms
(* (sqrt (+ x -1)) (sqrt x))

simplify349.0ms

Counts
18 → 27
Calls

18 calls. Slowest were:

182.0ms
(* (* (* (sqrt (+ x -1)) (sqrt (+ x -1))) (sqrt (+ x -1))) (* (* (sqrt x) (sqrt x)) (sqrt x)))
76.0ms
(* (sqrt (- (* x x) (* -1 -1))) (sqrt x))
37.0ms
(- x (+ (* 1/8 (/ 1 x)) 1/2))

prune302.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize19.0ms

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:

prune5.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize5.0ms

Local error

Found 1 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
1 → 24
Calls

1 calls. Slowest were:

9.0ms
(* (sqrt x) (sqrt (+ x -1)))

series24.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

24.0ms
(* (sqrt x) (sqrt (+ x -1)))

simplify430.0ms

Counts
18 → 27
Calls

18 calls. Slowest were:

209.0ms
(* (* (* (sqrt x) (sqrt x)) (sqrt x)) (* (* (sqrt (+ x -1)) (sqrt (+ x -1))) (sqrt (+ x -1))))
106.0ms
(sqrt (+ (* x x) (- (* -1 -1) (* x -1))))
64.0ms
(- x (+ (* 1/8 (/ 1 x)) 1/2))

prune492.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes78.0ms

Accuracy

0% (0.4b remaining)

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

bsearch2.0ms