Average Error: 14.3 → 0.3
Time: 8.5s
Precision: 64
Internal Precision: 128
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{-1}{\left(x + 1\right) \cdot x}\]

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. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019016 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Details

Time bar (total: 8.2s)Debug log

sample42.0ms

Algorithm
intervals
Results
256×(pre true 80)
131×(body real 80)
57×(body real 640)
35×(body real 160)
33×(body real 320)

simplify5.0ms

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

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 14.5b

localize11.0ms

Local error

Found 2 expressions with local error:

3.5b
(- (/ 1 (+ x 1)) (/ 1 x))
0.0b
(/ 1 (+ x 1))

rewrite7.0ms

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
5.0ms
(- (/ 1 (+ x 1)) (/ 1 x))
1.0ms
(/ 1 (+ x 1))

series36.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
21.0ms
(/ 1 (+ x 1))
15.0ms
(- (/ 1 (+ x 1)) (/ 1 x))

simplify381.0ms

Counts
21 → 42
Calls
21 calls:
Slowest
60.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
59.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
45.0ms
(- (+ (/ 1 (pow x 3)) (/ 1 x)) (/ 1 (pow x 2)))
30.0ms
(- (+ (/ 1 (pow x 3)) (/ 1 x)) (/ 1 (pow x 2)))
28.0ms
(/ (exp (/ 1 (+ x 1))) (exp (/ 1 x)))

prune317.0ms

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)

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
3.0ms
(* (+ x 1) x)
1.0ms
(/ -1 (* (+ x 1) x))

series27.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
19.0ms
(* (+ x 1) x)
9.0ms
(/ -1 (* (+ x 1) x))

simplify267.0ms

Counts
12 → 32
Calls
12 calls:
Slowest
74.0ms
(* (- (* x x) (* 1 1)) x)
52.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
37.0ms
(+ (* x x) (- (* 1 1) (* x 1)))
36.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
25.0ms
(- 1 (+ x (/ 1 x)))

prune230.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 2 expressions with local error:

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

rewrite7.0ms

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
5.0ms
(/ (/ -1 (+ x 1)) x)
1.0ms
(/ -1 (+ x 1))

series30.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
18.0ms
(/ (/ -1 (+ x 1)) x)
11.0ms
(/ -1 (+ x 1))

simplify407.0ms

Counts
15 → 38
Calls
15 calls:
Slowest
67.0ms
(- (/ 1 (pow x 2)) (+ (/ 1 (pow x 3)) (/ 1 x)))
61.0ms
(- (/ 1 (pow x 2)) (+ (/ 1 (pow x 3)) (/ 1 x)))
57.0ms
(/ -1 (- (* x x) (* 1 1)))
52.0ms
(- (/ 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))))

prune249.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

15.2b
(cbrt (* (+ x 1) x))
15.2b
(cbrt (* (+ x 1) x))
15.2b
(cbrt (* (+ x 1) x))
0.5b
(* (cbrt (* (+ x 1) x)) (cbrt (* (+ x 1) x)))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
20×associate-*l/
18×cbrt-div
9×flip-+
9×flip3-+
9×pow1
7×pow1/3
6×add-cube-cbrt
6×add-exp-log
6×*-un-lft-identity
6×add-sqr-sqrt
5×cbrt-prod
4×add-log-exp
4×associate-*l*
4×associate-*r*
4×add-cbrt-cube
4×frac-times
2×pow-prod-up
2×associate-*r/
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 65
Calls
4 calls:
Slowest
9.0ms
(* (cbrt (* (+ x 1) x)) (cbrt (* (+ x 1) x)))
2.0ms
(cbrt (* (+ x 1) x))
2.0ms
(cbrt (* (+ x 1) x))
2.0ms
(cbrt (* (+ x 1) x))

series245.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
70.0ms
(* (cbrt (* (+ x 1) x)) (cbrt (* (+ x 1) x)))
67.0ms
(cbrt (* (+ x 1) x))
63.0ms
(cbrt (* (+ x 1) x))
45.0ms
(cbrt (* (+ x 1) x))

simplify3.1s

Counts
49 → 77
Calls
49 calls:
Slowest
330.0ms
(* (* (+ x 1) x) (* (+ x 1) x))
300.0ms
(- (+ (* (pow (pow x 4) 1/3) (pow (cbrt -1) 2)) (* 2/3 (* (pow x 1/3) (pow (cbrt -1) 2)))) (* 1/9 (* (pow (/ 1 (pow x 2)) 1/3) (pow (cbrt -1) 2))))
272.0ms
(* (* (+ x 1) x) (* (+ x 1) x))
212.0ms
(- (+ (* 1/3 (/ (* (cbrt -1) (exp (* 1/3 (- (log -1) (* 2 (log (/ -1 x))))))) x)) (* (cbrt -1) (exp (* 1/3 (- (log -1) (* 2 (log (/ -1 x)))))))) (* 1/9 (/ (* (cbrt -1) (exp (* 1/3 (- (log -1) (* 2 (log (/ -1 x))))))) (pow x 2))))
194.0ms
(- (+ (* 1/3 (/ (* (cbrt -1) (exp (* 1/3 (- (log -1) (* 2 (log (/ -1 x))))))) x)) (* (cbrt -1) (exp (* 1/3 (- (log -1) (* 2 (log (/ -1 x)))))))) (* 1/9 (/ (* (cbrt -1) (exp (* 1/3 (- (log -1) (* 2 (log (/ -1 x))))))) (pow x 2))))

prune732.0ms

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.0b

regimes50.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch2.0ms

end0.0ms

sample2.0s

Algorithm
intervals
Results
8004×(pre true 80)
4205×(body real 80)
1895×(body real 640)
1224×(body real 320)
680×(body real 160)