Average Error: 29.0 → 0.2
Time: 15.0s
Precision: 64
Internal Precision: 128
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.0
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.0

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--28.9

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

  4. Taylor expanded around inf 0.2

    \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))

Details

Time bar (total: 11.5s)Debug log

start92.0ms

Algorithm
intervals

setup25.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 28.8b

localize10.0ms

Local error

Found 2 expressions with local error:

2.9b
(- (sqrt (+ x 1)) (sqrt x))
0.0b
(sqrt (+ x 1))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 57
Calls

2 calls. Slowest were:

6.0ms
(- (sqrt (+ x 1)) (sqrt x))
1.0ms
(sqrt (+ x 1))

series44.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

26.0ms
(- (sqrt (+ x 1)) (sqrt x))
18.0ms
(sqrt (+ x 1))

simplify903.0ms

Counts
59 → 63
Calls

59 calls. Slowest were:

69.0ms
(- (+ (* 1/2 x) 1) (* 1/8 (pow x 2)))
38.0ms
(fma (sqrt 1) (sqrt (+ x 1)) (- (* (sqrt x) 1)))
36.0ms
(fma (- (cbrt (sqrt x))) (* (cbrt (sqrt x)) (cbrt (sqrt x))) (* (cbrt (sqrt x)) (* (cbrt (sqrt x)) (cbrt (sqrt x)))))

prune618.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 28.0b

localize27.0ms

Local error

Found 4 expressions with local error:

2.4b
(- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x)))
0.5b
(* (sqrt x) (sqrt x))
0.3b
(* (sqrt (+ x 1)) (sqrt (+ x 1)))
0.0b
(/ (- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x))) (+ (sqrt (+ x 1)) (sqrt x)))

rewrite72.0ms

Algorithm
rewrite-expression-head
Counts
4 → 126
Calls

4 calls. Slowest were:

37.0ms
(/ (- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x))) (+ (sqrt (+ x 1)) (sqrt x)))
17.0ms
(- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x)))
9.0ms
(* (sqrt (+ x 1)) (sqrt (+ x 1)))

series72.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

38.0ms
(/ (- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x))) (+ (sqrt (+ x 1)) (sqrt x)))
13.0ms
(* (sqrt (+ x 1)) (sqrt (+ x 1)))
11.0ms
(* (sqrt x) (sqrt x))
10.0ms
(- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x)))

simplify3.3s

Counts
111 → 138
Calls

111 calls. Slowest were:

812.0ms
(- (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt (+ x 1)) (sqrt (+ x 1)))) (* (* (sqrt x) (sqrt x)) (* (sqrt x) (sqrt x))))
257.0ms
(* (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))) (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))))
181.0ms
(- (pow (* (sqrt (+ x 1)) (sqrt (+ x 1))) 3) (pow (* (sqrt x) (sqrt x)) 3))

prune1.6s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize3.0ms

Local error

Found 3 expressions with local error:

0.2b
(/ 1 (+ (sqrt (+ x 1)) (sqrt x)))
0.0b
(+ (sqrt (+ x 1)) (sqrt x))
0.0b
(sqrt (+ x 1))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 57
Calls

3 calls. Slowest were:

5.0ms
(+ (sqrt (+ x 1)) (sqrt x))
4.0ms
(/ 1 (+ (sqrt (+ x 1)) (sqrt x)))
1.0ms
(sqrt (+ x 1))

series74.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

33.0ms
(+ (sqrt (+ x 1)) (sqrt x))
29.0ms
(/ 1 (+ (sqrt (+ x 1)) (sqrt x)))
12.0ms
(sqrt (+ x 1))

simplify275.0ms

Counts
29 → 66
Calls

29 calls. Slowest were:

106.0ms
(+ (* (sqrt (+ x 1)) (sqrt (+ x 1))) (- (* (sqrt x) (sqrt x)) (* (sqrt (+ x 1)) (sqrt x))))
69.0ms
(- (+ (* 1/2 x) 1) (* 1/8 (pow x 2)))
35.0ms
(sqrt (- (* x x) (* 1 1)))

prune632.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

localize41.0ms

Local error

Found 4 expressions with local error:

0.3b
(cbrt (+ x 1))
0.3b
(cbrt (+ 1 x))
0.2b
(* (fabs (cbrt (+ 1 x))) (sqrt (cbrt (+ x 1))))
0.2b
(/ 1 (+ (* (fabs (cbrt (+ 1 x))) (sqrt (cbrt (+ x 1)))) (sqrt x)))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 77
Calls

4 calls. Slowest were:

13.0ms
(/ 1 (+ (* (fabs (cbrt (+ 1 x))) (sqrt (cbrt (+ x 1)))) (sqrt x)))
10.0ms
(* (fabs (cbrt (+ 1 x))) (sqrt (cbrt (+ x 1))))
1.0ms
(cbrt (+ x 1))

series280.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

106.0ms
(/ 1 (+ (* (fabs (cbrt (+ 1 x))) (sqrt (cbrt (+ x 1)))) (sqrt x)))
63.0ms
(* (fabs (cbrt (+ 1 x))) (sqrt (cbrt (+ x 1))))
58.0ms
(cbrt (+ x 1))
53.0ms
(cbrt (+ 1 x))

simplify2.1s

Counts
44 → 89
Calls

44 calls. Slowest were:

470.0ms
(- (+ (fabs (pow (+ x 1) 1/3)) (* 1/6 (* (fabs (pow (+ x 1) 1/3)) x))) (* 5/72 (* (fabs (pow (+ x 1) 1/3)) (pow x 2))))
288.0ms
(- (+ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (* 1/3 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) x))) (* 1/9 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (pow x 2))))
231.0ms
(- (+ (* 1/6 (* (fabs (pow (+ x 1) 1/3)) (pow (/ 1 (pow x 5)) 1/6))) (* (fabs (pow (+ x 1) 1/3)) (pow x 1/6))) (* 5/72 (* (fabs (pow (+ x 1) 1/3)) (pow (/ 1 (pow x 11)) 1/6))))

prune1.2s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 0.0b

regimes80.0ms

Accuracy

0% (0.1b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch2.0ms