Average Error: 29.8 → 0.2
Time: 13.0s
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

Original29.8
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.8

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

  3. Applied flip--29.6

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

  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 2018362 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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

Details

Time bar (total: 8.7s)Debug log

start98.0ms

Algorithm
intervals

setup24.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 27.8b

localize22.0ms

Local error

Found 2 expressions with local error:

1.6b
(- (sqrt (+ x 1)) (sqrt x))
0.0b
(sqrt (+ x 1))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
2 → 29
Calls

2 calls. Slowest were:

11.0ms
(- (sqrt (+ x 1)) (sqrt x))
1.0ms
(sqrt (+ x 1))

series47.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

30.0ms
(- (sqrt (+ x 1)) (sqrt x))
17.0ms
(sqrt (+ x 1))

simplify146.0ms

Counts
18 → 35
Calls

18 calls. Slowest were:

50.0ms
(- (+ (* 1/2 x) 1) (* 1/8 (pow x 2)))
21.0ms
(/ (exp (sqrt (+ x 1))) (exp (sqrt x)))
17.0ms
(sqrt (- (* x x) (* 1 1)))

prune314.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 27.3b

localize15.0ms

Local error

Found 4 expressions with local error:

1.5b
(- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x)))
0.5b
(* (sqrt x) (sqrt x))
0.2b
(* (sqrt (+ x 1)) (sqrt (+ x 1)))
0.0b
(/ (- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x))) (+ (sqrt (+ x 1)) (sqrt x)))

rewrite59.0ms

Algorithm
rewrite-expression-head
Counts
4 → 116
Calls

4 calls. Slowest were:

27.0ms
(/ (- (* (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)))
6.0ms
(* (sqrt (+ x 1)) (sqrt (+ x 1)))

series77.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

50.0ms
(/ (- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x))) (+ (sqrt (+ x 1)) (sqrt x)))
10.0ms
(- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x)))
9.0ms
(* (sqrt (+ x 1)) (sqrt (+ x 1)))
8.0ms
(* (sqrt x) (sqrt x))

simplify2.4s

Counts
108 → 128
Calls

108 calls. Slowest were:

486.0ms
(- (pow (* (sqrt (+ x 1)) (sqrt (+ x 1))) 3) (pow (* (sqrt x) (sqrt x)) 3))
421.0ms
(- (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt (+ x 1)) (sqrt (+ x 1)))) (* (* (sqrt x) (sqrt x)) (* (sqrt x) (sqrt x))))
163.0ms
(* (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))) (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))))

prune1.3s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.1b

localize6.0ms

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))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
3 → 45
Calls

3 calls. Slowest were:

7.0ms
(/ 1 (+ (sqrt (+ x 1)) (sqrt x)))
6.0ms
(+ (sqrt (+ x 1)) (sqrt x))
1.0ms
(sqrt (+ x 1))

series100.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

47.0ms
(/ 1 (+ (sqrt (+ x 1)) (sqrt x)))
35.0ms
(+ (sqrt (+ x 1)) (sqrt x))
18.0ms
(sqrt (+ x 1))

simplify285.0ms

Counts
24 → 54
Calls

24 calls. Slowest were:

157.0ms
(+ (* (sqrt (+ x 1)) (sqrt (+ x 1))) (- (* (sqrt x) (sqrt x)) (* (sqrt (+ x 1)) (sqrt x))))
40.0ms
(- (+ (* 1/2 x) 1) (* 1/8 (pow x 2)))
22.0ms
(sqrt (- (* x x) (* 1 1)))

prune575.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (sqrt (sqrt (+ x 1))) (sqrt (sqrt (+ x 1))))
0.2b
(/ 1 (+ (* (sqrt (sqrt (+ x 1))) (sqrt (sqrt (+ x 1)))) (sqrt x)))
0.1b
(sqrt (sqrt (+ x 1)))
0.1b
(sqrt (sqrt (+ x 1)))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 98
Calls

4 calls. Slowest were:

11.0ms
(/ 1 (+ (* (sqrt (sqrt (+ x 1))) (sqrt (sqrt (+ x 1)))) (sqrt x)))
6.0ms
(* (sqrt (sqrt (+ x 1))) (sqrt (sqrt (+ x 1))))
1.0ms
(sqrt (sqrt (+ x 1)))

series165.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

52.0ms
(sqrt (sqrt (+ x 1)))
47.0ms
(/ 1 (+ (* (sqrt (sqrt (+ x 1))) (sqrt (sqrt (+ x 1)))) (sqrt x)))
47.0ms
(sqrt (sqrt (+ x 1)))
19.0ms
(* (sqrt (sqrt (+ x 1))) (sqrt (sqrt (+ x 1))))

simplify1.6s

Counts
78 → 110
Calls

78 calls. Slowest were:

236.0ms
(- (+ (* 1/4 (/ (exp (* 1/4 (- (log -1) (log (/ -1 x))))) x)) (exp (* 1/4 (- (log -1) (log (/ -1 x)))))) (* 3/32 (/ (exp (* 1/4 (- (log -1) (log (/ -1 x))))) (pow x 2))))
168.0ms
(- (+ (* 1/4 (/ (exp (* 1/4 (- (log -1) (log (/ -1 x))))) x)) (exp (* 1/4 (- (log -1) (log (/ -1 x)))))) (* 3/32 (/ (exp (* 1/4 (- (log -1) (log (/ -1 x))))) (pow x 2))))
139.0ms
(+ (* (* (sqrt (sqrt (+ x 1))) (sqrt (sqrt (+ x 1)))) (* (sqrt (sqrt (+ x 1))) (sqrt (sqrt (+ x 1))))) (- (* (sqrt x) (sqrt x)) (* (* (sqrt (sqrt (+ x 1))) (sqrt (sqrt (+ x 1)))) (sqrt x))))

prune1.3s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

regimes102.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms