Average Error: 0.5 → 0.4
Time: 5.2s
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 2019010 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))

Details

Time bar (total: 4.9s)Debug log

sample193.0ms

Algorithm
intervals

simplify19.0ms

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

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize46.0ms

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

series34.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
34.0ms
(* (sqrt (- x 1)) (sqrt x))

simplify750.0ms

Counts
18 → 29
Calls
18 calls:
Slowest
456.0ms
(* (* (* (sqrt (- x 1)) (sqrt (- x 1))) (sqrt (- x 1))) (* (* (sqrt x) (sqrt x)) (sqrt x)))
129.0ms
(- x (+ (* 1/8 (/ 1 x)) 1/2))
57.0ms
(* (sqrt (- (* x x) (* 1 1))) (sqrt x))
20.0ms
(* (sqrt (- (pow x 3) (pow 1 3))) (sqrt x))
15.0ms
(* (- x 1) x)

prune202.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize17.0ms

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

prune1.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes51.0ms

Accuracy

0% (0.3b remaining)

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

bsearch1.0ms

end0.0ms

sample3.6s

Algorithm
intervals