Average Error: 14.3 → 0.1
Time: 6.7s
Precision: 64
Internal Precision: 128
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{-1}{x + 1}}{x}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.3

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

  3. Applied frac-sub13.7

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

  4. Simplified0.4

    \[\leadsto \frac{\color{blue}{-1}}{\left(x + 1\right) \cdot x}\]
  5. Using strategy rm

  6. Applied associate-/r*0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019021 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Details

Time bar (total: 6.2s)Debug log

sample40.0ms

Algorithm
intervals
Results
17.0ms68×body640valid
8.0ms134×body80valid
8.0ms41×body320valid
2.0ms13×body160valid

simplify5.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
4.0ms
(- (/ 1 (+ x 1)) (/ 1 x))

prune3.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 14.8b

localize14.0ms

Local error

Found 2 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
7×*-un-lft-identity
5×div-inv
5×add-sqr-sqrt
4×add-log-exp
4×distribute-lft-out--
3×add-cube-cbrt
3×add-exp-log
3×associate-/r*
3×pow1
2×associate-/r/
2×add-cbrt-cube
1×difference-of-squares
1×inv-pow
1×flip--
1×pow-flip
1×flip-+
1×frac-sub
1×diff-log
1×flip3--
1×flip3-+
1×frac-2neg
1×sub-neg
1×clear-num
1×rec-exp
Counts
2 → 36
Calls
2 calls:
Slowest
10.0ms
(- (/ 1 (+ x 1)) (/ 1 x))
2.0ms
(/ 1 (+ x 1))

series31.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
25.0ms
(- (/ 1 (+ x 1)) (/ 1 x))
6.0ms
(/ 1 (+ x 1))

simplify349.0ms

Counts
21 → 42
Calls
21 calls:
Slowest
66.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
45.0ms
(- 1 (+ x (/ 1 x)))
40.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
29.0ms
(- (* 1 x) (* (+ x 1) 1))
28.0ms
(- (+ (/ 1 (pow x 3)) (/ 1 x)) (/ 1 (pow x 2)))

prune274.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.5b

localize5.0ms

Local error

Found 2 expressions with local error:

0.5b
(/ -1 (* (+ x 1) x))
0.0b
(* (+ x 1) x)

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
4×associate-*l/
3×add-cube-cbrt
3×associate-*l*
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
2×flip-+
2×associate-/r/
2×add-exp-log
2×add-cbrt-cube
2×flip3-+
2×pow1
1×div-inv
1×*-commutative
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
2 → 26
Calls
2 calls:
Slowest
2.0ms
(* (+ x 1) x)
1.0ms
(/ -1 (* (+ x 1) x))

series28.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
19.0ms
(* (+ x 1) x)
8.0ms
(/ -1 (* (+ x 1) x))

simplify292.0ms

Counts
12 → 32
Calls
12 calls:
Slowest
92.0ms
(* (- (* x x) (* 1 1)) x)
47.0ms
(- 1 (+ x (/ 1 x)))
44.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
35.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
33.0ms
(+ (* x x) (- (* 1 1) (* x 1)))

prune178.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 2 expressions with local error:

0.1b
(/ (/ -1 (+ x 1)) x)
0.0b
(/ -1 (+ x 1))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
6×associate-/l*
4×add-cube-cbrt
4×associate-/r/
4×*-un-lft-identity
4×add-sqr-sqrt
3×div-inv
3×associate-/r*
2×add-log-exp
2×flip-+
2×add-exp-log
2×add-cbrt-cube
2×flip3-+
2×frac-2neg
2×clear-num
2×pow1
1×associate-/l/
Counts
2 → 32
Calls
2 calls:
Slowest
4.0ms
(/ (/ -1 (+ x 1)) x)
1.0ms
(/ -1 (+ x 1))

series23.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
13.0ms
(/ (/ -1 (+ x 1)) x)
10.0ms
(/ -1 (+ x 1))

simplify364.0ms

Counts
15 → 38
Calls
15 calls:
Slowest
68.0ms
(- (/ 1 (pow x 2)) (+ (/ 1 (pow x 3)) (/ 1 x)))
57.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
47.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
46.0ms
(/ -1 (- (* x x) (* 1 1)))
42.0ms
(- (/ 1 (pow x 2)) (+ (/ 1 (pow x 3)) (/ 1 x)))

prune272.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.5b
(/ -1 (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (* (cbrt (+ x 1)) x)))
0.3b
(cbrt (+ x 1))
0.3b
(cbrt (+ x 1))
0.3b
(cbrt (+ x 1))

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
60×cbrt-div
32×associate-*l/
30×flip-+
30×flip3-+
28×frac-times
26×associate-/r/
9×cbrt-prod
8×associate-*r/
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×pow1/3
1×div-inv
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
4 → 76
Calls
4 calls:
Slowest
7.0ms
(/ -1 (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (* (cbrt (+ x 1)) x)))
1.0ms
(cbrt (+ x 1))
1.0ms
(cbrt (+ x 1))
1.0ms
(cbrt (+ x 1))

series154.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
51.0ms
(cbrt (+ x 1))
46.0ms
(cbrt (+ x 1))
43.0ms
(cbrt (+ x 1))
14.0ms
(/ -1 (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (* (cbrt (+ x 1)) x)))

simplify1.9s

Counts
54 → 88
Calls
54 calls:
Slowest
210.0ms
(- (+ (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))))
197.0ms
(- (+ (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))))
158.0ms
(- (+ (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))))
76.0ms
(- (+ (* 1/3 x) 1) (* 1/9 (pow x 2)))
70.0ms
(- (+ (* 1/3 (pow (/ 1 (pow x 2)) 1/3)) (pow (/ 1 x) -1/3)) (* 1/9 (pow (/ 1 (pow x 5)) 1/3)))

prune843.0ms

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0b

regimes34.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch1.0ms

end0.0ms

sample1.3s

Algorithm
intervals
Results
539.0ms1822×body640valid
368.0ms4293×body80valid
268.0ms1277×body320valid
79.0ms612×body160valid