Average Error: 29.6 → 0.2
Time: 13.2s
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

Original29.6
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.6

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

  3. Applied flip--29.4

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

  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 2019016 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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

Details

Time bar (total: 12.3s)Debug log

sample87.0ms

Algorithm
intervals
Results
573×(pre true 80)
316×(body nan 80)
138×(body real 80)
60×(body real 1280)
39×(body real 640)
12×(body real 320)
8×(body real 160)

simplify8.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
8.0ms
(- (sqrt (+ x 1)) (sqrt x))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 28.5b

localize21.0ms

Local error

Found 2 expressions with local error:

2.4b
(- (sqrt (+ x 1)) (sqrt x))
0.0b
(sqrt (+ x 1))

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
8×add-sqr-sqrt
5×*-un-lft-identity
4×add-log-exp
4×sqrt-prod
3×add-cube-cbrt
2×difference-of-squares
2×add-exp-log
2×sqrt-div
2×add-cbrt-cube
2×pow1
1×distribute-lft-out--
1×flip--
1×flip-+
1×diff-log
1×pow1/2
1×flip3--
1×flip3-+
1×sub-neg
1×rem-sqrt-square
Counts
2 → 29
Calls
2 calls:
Slowest
11.0ms
(- (sqrt (+ x 1)) (sqrt x))
1.0ms
(sqrt (+ x 1))

series36.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
26.0ms
(- (sqrt (+ x 1)) (sqrt x))
10.0ms
(sqrt (+ x 1))

simplify118.0ms

Counts
18 → 35
Calls
18 calls:
Slowest
52.0ms
(- (+ (* 1/2 x) 1) (* 1/8 (pow x 2)))
19.0ms
(/ (exp (sqrt (+ x 1))) (exp (sqrt x)))
14.0ms
(sqrt (- (* x x) (* 1 1)))
4.0ms
(- (sqrt (sqrt (+ x 1))) (sqrt (sqrt x)))
4.0ms
(- (sqrt (sqrt (+ x 1))) (sqrt (sqrt x)))

prune342.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 27.8b

localize31.0ms

Local error

Found 4 expressions with local error:

1.7b
(- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x)))
0.5b
(* (sqrt x) (sqrt x))
0.3b
(* (sqrt (+ x 1)) (sqrt (+ x 1)))
0.1b
(/ (- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x))) (+ (sqrt (+ x 1)) (sqrt x)))

rewrite80.0ms

Algorithm
rewrite-expression-head
Rules
30×*-un-lft-identity
20×add-cube-cbrt
20×add-sqr-sqrt
16×times-frac
14×pow1
12×sqrt-div
10×add-exp-log
10×add-cbrt-cube
9×associate-*l*
9×associate-*r*
8×pow1/2
7×flip-+
7×flip3-+
6×difference-of-squares
6×add-log-exp
6×sqrt-prod
5×distribute-lft-out
4×associate-/l*
4×pow-prod-up
4×pow-prod-down
4×associate-/r*
4×frac-times
2×associate-/l/
2×flip--
2×cbrt-unprod
2×*-commutative
2×associate-/r/
2×associate-*r/
2×prod-exp
2×associate-*l/
2×sqrt-unprod
2×flip3--
2×pow-plus
2×rem-square-sqrt
2×pow2
1×div-inv
1×div-exp
1×div-sub
1×diff-log
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
4 → 116
Calls
4 calls:
Slowest
41.0ms
(/ (- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x))) (+ (sqrt (+ x 1)) (sqrt x)))
19.0ms
(- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x)))
10.0ms
(* (sqrt (+ x 1)) (sqrt (+ x 1)))
7.0ms
(* (sqrt x) (sqrt x))

series76.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
36.0ms
(/ (- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x))) (+ (sqrt (+ x 1)) (sqrt x)))
15.0ms
(* (sqrt (+ x 1)) (sqrt (+ x 1)))
12.0ms
(* (sqrt x) (sqrt x))
12.0ms
(- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x)))

simplify2.4s

Counts
108 → 128
Calls
108 calls:
Slowest
472.0ms
(- (pow (* (sqrt (+ x 1)) (sqrt (+ x 1))) 3) (pow (* (sqrt x) (sqrt x)) 3))
416.0ms
(- (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt (+ x 1)) (sqrt (+ x 1)))) (* (* (sqrt x) (sqrt x)) (* (sqrt x) (sqrt x))))
178.0ms
(* (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))) (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))))
98.0ms
(+ (* (sqrt (+ x 1)) (sqrt (+ x 1))) (- (* (sqrt x) (sqrt x)) (* (sqrt (+ x 1)) (sqrt x))))
82.0ms
(/ (* (* (- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x))) (- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x)))) (- (* (sqrt (+ x 1)) (sqrt (+ x 1))) (* (sqrt x) (sqrt x)))) (* (* (+ (sqrt (+ x 1)) (sqrt x)) (+ (sqrt (+ x 1)) (sqrt x))) (+ (sqrt (+ x 1)) (sqrt x))))

prune1.2s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.1b

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))

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
9×*-un-lft-identity
6×add-sqr-sqrt
5×add-log-exp
5×add-cube-cbrt
4×add-exp-log
4×associate-/r*
4×pow1
3×flip-+
3×sqrt-prod
3×add-cbrt-cube
3×flip3-+
2×distribute-lft-out
2×associate-/r/
2×sqrt-div
1×inv-pow
1×pow-flip
1×div-inv
1×pow1/2
1×frac-2neg
1×sum-log
1×clear-num
1×rec-exp
1×rem-sqrt-square
1×+-commutative
Counts
3 → 45
Calls
3 calls:
Slowest
5.0ms
(/ 1 (+ (sqrt (+ x 1)) (sqrt x)))
5.0ms
(+ (sqrt (+ x 1)) (sqrt x))
1.0ms
(sqrt (+ x 1))

series72.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
42.0ms
(/ 1 (+ (sqrt (+ x 1)) (sqrt x)))
20.0ms
(+ (sqrt (+ x 1)) (sqrt x))
10.0ms
(sqrt (+ x 1))

simplify317.0ms

Counts
24 → 54
Calls
24 calls:
Slowest
150.0ms
(+ (* (sqrt (+ x 1)) (sqrt (+ x 1))) (- (* (sqrt x) (sqrt x)) (* (sqrt (+ x 1)) (sqrt x))))
81.0ms
(- (+ (* 1/2 x) 1) (* 1/8 (pow x 2)))
29.0ms
(sqrt (- (* x x) (* 1 1)))
10.0ms
(* (exp (sqrt (+ x 1))) (exp (sqrt x)))
8.0ms
(- (sqrt (+ x 1)) (sqrt x))

prune450.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.1b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite50.0ms

Algorithm
rewrite-expression-head
Rules
50×*-un-lft-identity
42×times-frac
40×sqrt-prod
32×add-sqr-sqrt
30×add-cube-cbrt
24×sqrt-div
20×associate-/r/
14×associate-/r*
12×flip-+
12×flip3-+
10×distribute-lft-out
10×div-inv
9×add-exp-log
8×pow1
6×associate-/l*
6×add-cbrt-cube
5×pow1/2
4×add-log-exp
4×pow-flip
3×pow-div
2×inv-pow
2×div-exp
2×frac-2neg
2×clear-num
2×rec-exp
2×rem-sqrt-square
1×associate-/l/
1×cbrt-undiv
Counts
4 → 127
Calls
4 calls:
Slowest
20.0ms
(/ (/ 1 (sqrt (+ (sqrt (+ x 1)) (sqrt x)))) (sqrt (+ (sqrt (+ x 1)) (sqrt x))))
10.0ms
(sqrt (+ (sqrt (+ x 1)) (sqrt x)))
10.0ms
(sqrt (+ (sqrt (+ x 1)) (sqrt x)))
8.0ms
(/ 1 (sqrt (+ (sqrt (+ x 1)) (sqrt x))))

series229.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
75.0ms
(/ 1 (sqrt (+ (sqrt (+ x 1)) (sqrt x))))
57.0ms
(sqrt (+ (sqrt (+ x 1)) (sqrt x)))
48.0ms
(sqrt (+ (sqrt (+ x 1)) (sqrt x)))
48.0ms
(/ (/ 1 (sqrt (+ (sqrt (+ x 1)) (sqrt x)))) (sqrt (+ (sqrt (+ x 1)) (sqrt x))))

simplify2.7s

Counts
143 → 139
Calls
143 calls:
Slowest
456.0ms
(/ (* (* (/ 1 (sqrt (+ (sqrt (+ x 1)) (sqrt x)))) (/ 1 (sqrt (+ (sqrt (+ x 1)) (sqrt x))))) (/ 1 (sqrt (+ (sqrt (+ x 1)) (sqrt x))))) (* (* (sqrt (+ (sqrt (+ x 1)) (sqrt x))) (sqrt (+ (sqrt (+ x 1)) (sqrt x)))) (sqrt (+ (sqrt (+ x 1)) (sqrt x)))))
163.0ms
(/ (sqrt (+ (* (sqrt (+ x 1)) (sqrt (+ x 1))) (- (* (sqrt x) (sqrt x)) (* (sqrt (+ x 1)) (sqrt x))))) (sqrt (+ (sqrt (+ x 1)) (sqrt x))))
135.0ms
(/ (sqrt (+ (* (sqrt (+ x 1)) (sqrt (+ x 1))) (- (* (sqrt x) (sqrt x)) (* (sqrt (+ x 1)) (sqrt x))))) (sqrt (+ (sqrt (+ x 1)) (sqrt x))))
130.0ms
(/ (sqrt (+ (* (sqrt (+ x 1)) (sqrt (+ x 1))) (- (* (sqrt x) (sqrt x)) (* (sqrt (+ x 1)) (sqrt x))))) (cbrt (sqrt (+ (sqrt (+ x 1)) (sqrt x)))))
128.0ms
(/ (sqrt (+ (* (sqrt (+ x 1)) (sqrt (+ x 1))) (- (* (sqrt x) (sqrt x)) (* (sqrt (+ x 1)) (sqrt x))))) (sqrt (+ (sqrt (+ x 1)) (sqrt x))))

prune1.3s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

regimes74.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms

end0.0ms

sample2.5s

Algorithm
intervals
Results
16094×(pre true 80)
8093×(body nan 80)
4126×(body real 80)
1738×(body real 1280)
1214×(body real 640)
608×(body real 320)
315×(body real 160)