Average Error: 14.8 → 0.1
Time: 13.9s
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 14.8

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

  3. Applied frac-sub14.1

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

  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 1042149663 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Details

Time bar (total: 13.1s)Debug log

sample92.0ms

Algorithm
intervals
Results
256×(pre true 80)
132×(body real 80)
67×(body real 640)
45×(body real 320)
12×(body real 160)

simplify11.0ms

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

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 14.7b

localize30.0ms

Local error

Found 2 expressions with local error:

2.6b
(- (/ 1 (+ x 1)) (/ 1 x))
0.0b
(/ 1 (+ x 1))

rewrite14.0ms

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
11.0ms
(- (/ 1 (+ x 1)) (/ 1 x))
2.0ms
(/ 1 (+ x 1))

series45.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
23.0ms
(/ 1 (+ x 1))
21.0ms
(- (/ 1 (+ x 1)) (/ 1 x))

simplify626.0ms

Counts
21 → 42
Calls
21 calls:
Slowest
88.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
86.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
65.0ms
(- (+ (/ 1 (pow x 3)) (/ 1 x)) (/ 1 (pow x 2)))
64.0ms
(- (+ (/ 1 (pow x 3)) (/ 1 x)) (/ 1 (pow x 2)))
57.0ms
(- 1 (+ x (/ 1 x)))

prune459.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.2b

localize15.0ms

Local error

Found 2 expressions with local error:

0.2b
(/ -1 (* (+ x 1) x))
0.0b
(* (+ x 1) x)

rewrite8.0ms

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))

series55.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
37.0ms
(* (+ x 1) x)
17.0ms
(/ -1 (* (+ x 1) x))

simplify496.0ms

Counts
12 → 32
Calls
12 calls:
Slowest
141.0ms
(* (- (* x x) (* 1 1)) x)
90.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
86.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
58.0ms
(- 1 (+ x (/ 1 x)))
51.0ms
(+ (* x x) (- (* 1 1) (* x 1)))

prune360.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 2 expressions with local error:

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

rewrite8.0ms

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
7.0ms
(/ (/ -1 (+ x 1)) x)
1.0ms
(/ -1 (+ x 1))

series31.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
17.0ms
(/ (/ -1 (+ x 1)) x)
13.0ms
(/ -1 (+ x 1))

simplify600.0ms

Counts
15 → 38
Calls
15 calls:
Slowest
88.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
88.0ms
(- (/ 1 (pow x 2)) (+ (/ 1 (pow x 3)) (/ 1 x)))
86.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
83.0ms
(- (/ 1 (pow x 2)) (+ (/ 1 (pow x 3)) (/ 1 x)))
63.0ms
(/ -1 (- (* x x) (* 1 1)))

prune416.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

15.9b
(cbrt (/ (/ -1 (+ x 1)) x))
15.9b
(cbrt (/ (/ -1 (+ x 1)) x))
15.9b
(cbrt (/ (/ -1 (+ x 1)) x))
0.4b
(* (cbrt (/ (/ -1 (+ x 1)) x)) (cbrt (/ (/ -1 (+ x 1)) x)))

rewrite19.0ms

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

series400.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
113.0ms
(* (cbrt (/ (/ -1 (+ x 1)) x)) (cbrt (/ (/ -1 (+ x 1)) x)))
99.0ms
(cbrt (/ (/ -1 (+ x 1)) x))
95.0ms
(cbrt (/ (/ -1 (+ x 1)) x))
93.0ms
(cbrt (/ (/ -1 (+ x 1)) x))

simplify5.5s

Counts
53 → 84
Calls
53 calls:
Slowest
499.0ms
(- (+ (* 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))))
492.0ms
(- (+ (* 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))))
398.0ms
(* (/ (/ -1 (+ x 1)) x) (/ (/ -1 (+ x 1)) x))
394.0ms
(* (/ (/ -1 (+ x 1)) x) (/ (/ -1 (+ x 1)) x))
366.0ms
(- (+ (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)))

prune1.2s

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0b

regimes62.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms

end0.0ms

sample2.6s

Algorithm
intervals
Results
8000×(pre true 80)
4168×(body real 80)
1980×(body real 640)
1245×(body real 320)
607×(body real 160)