Average Error: 19.9 → 0.4
Time: 30.0s
Precision: 64
Internal Precision: 128
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{x + 1}}{\frac{x}{\sqrt{x}} + \frac{x}{\sqrt{x + 1}}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.9

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

  3. Applied flip--20.0

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
  4. Using strategy rm

  5. Applied frac-times25.0

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

  6. Applied frac-times20.1

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

  7. Applied frac-sub19.8

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

  8. Simplified5.9

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

  9. Simplified5.7

    \[\leadsto \frac{\frac{1}{\color{blue}{x \cdot x + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
  10. Using strategy rm

  11. Applied div-inv5.7

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

  12. Applied div-inv5.7

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

  13. Applied distribute-lft-out5.7

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

  14. Applied *-un-lft-identity5.7

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

  15. Applied times-frac5.7

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

  16. Simplified5.7

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

  17. Simplified0.4

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

  18. Final simplification0.4

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

Reproduce

herbie shell --seed 2019010 
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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

Details

Time bar (total: 29.4s)Debug log

sample114.0ms

Algorithm
intervals

simplify6.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
6.0ms
(- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1))))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 19.2b

localize29.0ms

Local error

Found 4 expressions with local error:

3.3b
(- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1))))
0.3b
(/ 1 (sqrt x))
0.1b
(/ 1 (sqrt (+ x 1)))
0.0b
(sqrt (+ x 1))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
12×*-un-lft-identity
11×add-sqr-sqrt
9×associate-/r*
8×add-cube-cbrt
6×add-log-exp
6×div-inv
6×sqrt-prod
6×add-exp-log
6×pow1
4×distribute-lft-out--
4×pow-flip
4×sqrt-div
4×add-cbrt-cube
3×pow1/2
2×inv-pow
2×flip-+
2×associate-/r/
2×flip3-+
2×frac-2neg
2×clear-num
2×rec-exp
1×difference-of-squares
1×flip--
1×frac-sub
1×diff-log
1×flip3--
1×sub-neg
1×rem-sqrt-square
Counts
4 → 71
Calls
4 calls:
Slowest
5.0ms
(- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1))))
1.0ms
(/ 1 (sqrt (+ x 1)))
1.0ms
(/ 1 (sqrt x))
1.0ms
(sqrt (+ x 1))

series95.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
39.0ms
(- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1))))
21.0ms
(/ 1 (sqrt (+ x 1)))
19.0ms
(/ 1 (sqrt x))
17.0ms
(sqrt (+ x 1))

simplify319.0ms

Counts
42 → 83
Calls
42 calls:
Slowest
64.0ms
(- (+ (* 3/8 (pow x 2)) 1) (* 1/2 x))
61.0ms
(- (+ (* 1/2 x) 1) (* 1/8 (pow x 2)))
37.0ms
(/ (exp (/ 1 (sqrt x))) (exp (/ 1 (sqrt (+ x 1)))))
30.0ms
(- (* 1 (sqrt (+ x 1))) (* (sqrt x) 1))
23.0ms
(sqrt (+ (* x x) (- (* 1 1) (* x 1))))

prune652.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 18.6b

localize30.0ms

Local error

Found 4 expressions with local error:

3.3b
(- (* (/ 1 (sqrt x)) (/ 1 (sqrt x))) (* (/ 1 (sqrt (+ x 1))) (/ 1 (sqrt (+ x 1)))))
0.5b
(* (/ 1 (sqrt x)) (/ 1 (sqrt x)))
0.3b
(/ (- (* (/ 1 (sqrt x)) (/ 1 (sqrt x))) (* (/ 1 (sqrt (+ x 1))) (/ 1 (sqrt (+ x 1))))) (+ (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))
0.3b
(/ 1 (sqrt x))

rewrite75.0ms

Algorithm
rewrite-expression-head
Rules
40×*-un-lft-identity
32×frac-sub
28×times-frac
24×div-inv
20×distribute-lft-out
20×add-cube-cbrt
20×add-sqr-sqrt
18×associate-/l/
17×un-div-inv
17×associate-*r/
17×associate-*l/
17×frac-times
16×pow1
15×add-exp-log
12×pow-flip
10×associate-/r*
9×difference-of-squares
8×add-cbrt-cube
7×inv-pow
6×add-log-exp
6×pow-prod-up
5×pow1/2
5×rec-exp
4×associate-/l*
4×prod-exp
4×pow-prod-down
4×associate-*l*
4×associate-*r*
3×associate-/r/
2×flip--
2×flip3--
2×frac-2neg
2×clear-num
1×flip-+
1×cbrt-unprod
1×*-commutative
1×frac-add
1×div-exp
1×div-sub
1×diff-log
1×pow-plus
1×flip3-+
1×sub-neg
1×cbrt-undiv
1×pow2
Counts
4 → 156
Calls
4 calls:
Slowest
43.0ms
(/ (- (* (/ 1 (sqrt x)) (/ 1 (sqrt x))) (* (/ 1 (sqrt (+ x 1))) (/ 1 (sqrt (+ x 1))))) (+ (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))
18.0ms
(- (* (/ 1 (sqrt x)) (/ 1 (sqrt x))) (* (/ 1 (sqrt (+ x 1))) (/ 1 (sqrt (+ x 1)))))
9.0ms
(* (/ 1 (sqrt x)) (/ 1 (sqrt x)))
1.0ms
(/ 1 (sqrt x))

series107.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
57.0ms
(/ (- (* (/ 1 (sqrt x)) (/ 1 (sqrt x))) (* (/ 1 (sqrt (+ x 1))) (/ 1 (sqrt (+ x 1))))) (+ (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))
27.0ms
(- (* (/ 1 (sqrt x)) (/ 1 (sqrt x))) (* (/ 1 (sqrt (+ x 1))) (/ 1 (sqrt (+ x 1)))))
11.0ms
(/ 1 (sqrt x))
11.0ms
(* (/ 1 (sqrt x)) (/ 1 (sqrt x)))

simplify15.0s

Counts
176 → 168
Calls
176 calls:
Slowest
611.0ms
(/ (sqrt (- (* (/ 1 (sqrt x)) (/ 1 (sqrt x))) (* (/ 1 (sqrt (+ x 1))) (/ 1 (sqrt (+ x 1)))))) (sqrt (+ (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1))))))
408.0ms
(/ (- (* (/ 1 (sqrt x)) (/ 1 (sqrt x))) (* (/ 1 (sqrt (+ x 1))) (/ 1 (sqrt (+ x 1))))) (+ (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))
389.0ms
(/ (- (* (/ 1 (sqrt x)) (/ 1 (sqrt x))) (* (/ 1 (sqrt (+ x 1))) (/ 1 (sqrt (+ x 1))))) (+ (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))
388.0ms
(/ (- (* (/ 1 (sqrt x)) (/ 1 (sqrt x))) (* (/ 1 (sqrt (+ x 1))) (/ 1 (sqrt (+ x 1))))) (+ (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))
373.0ms
(/ (- (* (/ 1 (sqrt x)) (/ 1 (sqrt x))) (* (/ 1 (sqrt (+ x 1))) (/ 1 (sqrt (+ x 1))))) (+ (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))

prune1.0s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 4.7b

localize10.0ms

Local error

Found 4 expressions with local error:

4.6b
(/ (/ 1 (+ (* x x) x)) (+ (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))
0.3b
(/ 1 (+ (* x x) x))
0.3b
(/ 1 (sqrt x))
0.1b
(/ 1 (sqrt (+ x 1)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
51×*-un-lft-identity
42×times-frac
40×div-inv
28×distribute-lft-out
23×add-cube-cbrt
23×associate-/r/
23×add-sqr-sqrt
20×associate-/r*
11×flip-+
11×add-exp-log
11×flip3-+
7×pow1
6×associate-/l*
6×add-cbrt-cube
5×pow-flip
4×add-log-exp
4×frac-2neg
4×clear-num
4×rec-exp
3×inv-pow
3×sqrt-prod
2×div-exp
2×sqrt-div
2×pow1/2
1×associate-/l/
1×frac-add
1×distribute-lft1-in
1×cbrt-undiv
Counts
4 → 130
Calls
4 calls:
Slowest
10.0ms
(/ (/ 1 (+ (* x x) x)) (+ (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))
2.0ms
(/ 1 (+ (* x x) x))
1.0ms
(/ 1 (sqrt (+ x 1)))
1.0ms
(/ 1 (sqrt x))

series86.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
54.0ms
(/ (/ 1 (+ (* x x) x)) (+ (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))
12.0ms
(/ 1 (sqrt (+ x 1)))
11.0ms
(/ 1 (sqrt x))
9.0ms
(/ 1 (+ (* x x) x))

simplify5.4s

Counts
134 → 142
Calls
134 calls:
Slowest
324.0ms
(+ (* (* x x) (* x x)) (- (* x x) (* (* x x) x)))
212.0ms
(/ (/ 1 (- (* (* x x) (* x x)) (* x x))) 1)
210.0ms
(/ (/ 1 (- (* (* x x) (* x x)) (* x x))) 1)
209.0ms
(/ (/ 1 (- (* (* x x) (* x x)) (* x x))) 1)
209.0ms
(/ (/ 1 (- (* (* x x) (* x x)) (* x x))) 1)

prune867.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

0.5b
(/ x (sqrt x))
0.3b
(/ (/ 1 (+ x 1)) (+ (/ x (sqrt x)) (/ x (sqrt (+ x 1)))))
0.2b
(/ x (sqrt (+ x 1)))
0.0b
(+ (/ x (sqrt x)) (/ x (sqrt (+ x 1))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
36×*-un-lft-identity
30×times-frac
25×div-inv
20×add-cube-cbrt
20×add-sqr-sqrt
17×associate-/r/
16×distribute-lft-out
14×associate-/r*
9×flip-+
9×flip3-+
8×add-exp-log
6×add-log-exp
6×associate-/l*
6×add-cbrt-cube
4×pow1
3×sqrt-prod
3×frac-2neg
3×clear-num
2×frac-add
2×div-exp
2×sqrt-div
1×associate-/l/
1×sum-log
1×rec-exp
1×+-commutative
1×cbrt-undiv
Counts
4 → 103
Calls
4 calls:
Slowest
9.0ms
(/ (/ 1 (+ x 1)) (+ (/ x (sqrt x)) (/ x (sqrt (+ x 1)))))
5.0ms
(+ (/ x (sqrt x)) (/ x (sqrt (+ x 1))))
1.0ms
(/ x (sqrt (+ x 1)))
1.0ms
(/ x (sqrt x))

series133.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
58.0ms
(/ (/ 1 (+ x 1)) (+ (/ x (sqrt x)) (/ x (sqrt (+ x 1)))))
37.0ms
(+ (/ x (sqrt x)) (/ x (sqrt (+ x 1))))
21.0ms
(/ x (sqrt (+ x 1)))
18.0ms
(/ x (sqrt x))

simplify2.5s

Counts
103 → 115
Calls
103 calls:
Slowest
172.0ms
(+ (* (/ x (sqrt x)) (/ x (sqrt x))) (- (* (/ x (sqrt (+ x 1))) (/ x (sqrt (+ x 1)))) (* (/ x (sqrt x)) (/ x (sqrt (+ x 1))))))
168.0ms
(/ (/ 1 (+ x 1)) (+ (/ x (sqrt x)) (/ x (sqrt (+ x 1)))))
167.0ms
(/ (/ 1 (+ x 1)) (+ (/ x (sqrt x)) (/ x (sqrt (+ x 1)))))
167.0ms
(/ (/ 1 (+ x 1)) (+ (/ x (sqrt x)) (/ x (sqrt (+ x 1)))))
165.0ms
(/ (/ 1 (+ x 1)) (+ (/ x (sqrt x)) (/ x (sqrt (+ x 1)))))

prune644.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0b

regimes40.0ms

Accuracy

0% (0.3b remaining)

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

bsearch1.0ms

end0.0ms

sample2.3s

Algorithm
intervals