Average Error: 14.3 → 0.1
Time: 8.5s
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-sub13.6

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

  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 2019022 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Details

Time bar (total: 8.1s)Debug log

sample32.0ms

Algorithm
intervals
Results
10.0ms56×body640valid
7.0ms135×body80valid
6.0ms45×body320valid
2.0ms21×body160valid

simplify5.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
5.0ms
(- (/ 1 (+ x 1)) (/ 1 x))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 15.1b

localize23.0ms

Local error

Found 2 expressions with local error:

4.1b
(- (/ 1 (+ x 1)) (/ 1 x))
0.0b
(/ 1 (+ x 1))

rewrite15.0ms

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
12.0ms
(- (/ 1 (+ x 1)) (/ 1 x))
2.0ms
(/ 1 (+ x 1))

series40.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
22.0ms
(/ 1 (+ x 1))
18.0ms
(- (/ 1 (+ x 1)) (/ 1 x))

simplify391.0ms

Counts
21 → 42
Calls
21 calls:
Slowest
49.0ms
(- (+ (/ 1 (pow x 3)) (/ 1 x)) (/ 1 (pow x 2)))
45.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
44.0ms
(- (* 1 x) (* (+ x 1) 1))
43.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
42.0ms
(- (+ (/ 1 (pow x 3)) (/ 1 x)) (/ 1 (pow x 2)))

prune261.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

localize5.0ms

Local error

Found 2 expressions with local error:

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

rewrite7.0ms

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
5.0ms
(* (+ x 1) x)
2.0ms
(/ -1 (* (+ x 1) x))

series34.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
19.0ms
(* (+ x 1) x)
14.0ms
(/ -1 (* (+ x 1) x))

simplify288.0ms

Counts
12 → 32
Calls
12 calls:
Slowest
81.0ms
(* (- (* x x) (* 1 1)) x)
68.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
36.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
34.0ms
(+ (* x x) (- (* 1 1) (* x 1)))
24.0ms
(- 1 (+ x (/ 1 x)))

prune251.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 2 expressions with local error:

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

rewrite4.0ms

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
3.0ms
(/ (/ -1 (+ x 1)) x)
0.0ms
(/ -1 (+ x 1))

series16.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
9.0ms
(/ (/ -1 (+ x 1)) x)
7.0ms
(/ -1 (+ x 1))

simplify359.0ms

Counts
15 → 38
Calls
15 calls:
Slowest
53.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
53.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
46.0ms
(- 1 (+ x (/ 1 x)))
41.0ms
(- (/ 1 (pow x 2)) (+ (/ 1 (pow x 3)) (/ 1 x)))
41.0ms
(+ (* x x) (- (* 1 1) (* x 1)))

prune270.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

14.4b
(cbrt (/ (/ -1 (+ x 1)) x))
14.4b
(cbrt (/ (/ -1 (+ x 1)) x))
14.4b
(cbrt (/ (/ -1 (+ x 1)) x))
0.5b
(* (cbrt (/ (/ -1 (+ x 1)) x)) (cbrt (/ (/ -1 (+ x 1)) x)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
20×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×pow1
7×associate-*l*
7×associate-*r*
7×pow1/3
7×cbrt-div
6×add-exp-log
5×div-inv
4×add-log-exp
4×add-cbrt-cube
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-plus
1×frac-times
1×pow2
Counts
4 → 72
Calls
4 calls:
Slowest
5.0ms
(* (cbrt (/ (/ -1 (+ x 1)) x)) (cbrt (/ (/ -1 (+ x 1)) x)))
1.0ms
(cbrt (/ (/ -1 (+ x 1)) x))
1.0ms
(cbrt (/ (/ -1 (+ x 1)) x))
1.0ms
(cbrt (/ (/ -1 (+ x 1)) x))

series236.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
71.0ms
(cbrt (/ (/ -1 (+ x 1)) x))
70.0ms
(* (cbrt (/ (/ -1 (+ x 1)) x)) (cbrt (/ (/ -1 (+ x 1)) x)))
51.0ms
(cbrt (/ (/ -1 (+ x 1)) x))
44.0ms
(cbrt (/ (/ -1 (+ x 1)) x))

simplify3.3s

Counts
53 → 84
Calls
53 calls:
Slowest
282.0ms
(* (/ (/ -1 (+ x 1)) x) (/ (/ -1 (+ x 1)) x))
278.0ms
(- (+ (* 5/9 (* (pow (/ 1 (pow x 10)) 1/3) (pow (cbrt -1) 2))) (* (pow (/ 1 (pow x 4)) 1/3) (pow (cbrt -1) 2))) (* 2/3 (* (pow (/ 1 (pow x 7)) 1/3) (pow (cbrt -1) 2))))
257.0ms
(- (+ (exp (* 1/3 (+ (* 2 (log (/ -1 x))) (log -1)))) (* 2/9 (/ (exp (* 1/3 (+ (* 2 (log (/ -1 x))) (log -1)))) (pow x 2)))) (* 1/3 (/ (exp (* 1/3 (+ (* 2 (log (/ -1 x))) (log -1)))) x)))
247.0ms
(- (+ (* 5/9 (* (pow (pow x 4) 1/3) (pow (cbrt -1) 2))) (* (pow (/ 1 (pow x 2)) 1/3) (pow (cbrt -1) 2))) (* 2/3 (* (pow x 1/3) (pow (cbrt -1) 2))))
242.0ms
(* (/ (/ -1 (+ x 1)) x) (/ (/ -1 (+ x 1)) x))

prune931.0ms

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0.0b

regimes52.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms

end0.0ms

sample1.5s

Algorithm
intervals
Results
648.0ms1845×body640valid
356.0ms1284×body320valid
337.0ms4232×body80valid
91.0ms641×body160valid