Average Error: 9.9 → 0.1
Time: 39.6s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.9

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  2. Using strategy rm

  3. Applied frac-sub25.6

    \[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]

  4. Applied frac-add25.0

    \[\leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]

  5. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
  6. Using strategy rm

  7. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}}\]

  8. Final simplification0.1

    \[\leadsto \frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}\]

Reproduce

herbie shell --seed 2019018 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))

Details

Time bar (total: 39.3s)Debug log

sample87.0ms

Algorithm
intervals
Results
36.0ms72×body640valid
16.0ms124×body80valid
11.0ms17×body1280valid
9.0ms25×body320valid
5.0ms18×body160valid

simplify50.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
50.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 14.2b

localize34.0ms

Local error

Found 2 expressions with local error:

3.2b
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
0.0b
(- (/ 1 (+ x 1)) (/ 2 x))

rewrite27.0ms

Algorithm
rewrite-expression-head
Rules
16×*-un-lft-identity
9×add-log-exp
6×distribute-lft-out--
6×distribute-lft-out
6×div-inv
4×add-sqr-sqrt
3×frac-add
2×flip--
2×add-cube-cbrt
2×frac-sub
2×add-exp-log
2×diff-log
2×add-cbrt-cube
2×flip3--
2×sub-neg
2×sum-log
2×associate-+l+
2×pow1
1×difference-of-squares
1×flip-+
1×associate-+l-
1×flip3-+
1×+-commutative
Counts
2 → 40
Calls
2 calls:
Slowest
21.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
5.0ms
(- (/ 1 (+ x 1)) (/ 2 x))

series61.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
50.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
11.0ms
(- (/ 1 (+ x 1)) (/ 2 x))

simplify2.9s

Counts
30 → 46
Calls
30 calls:
Slowest
471.0ms
(* (+ (/ 1 (+ x 1)) (/ 2 x)) (- x 1))
380.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))
289.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))
241.0ms
(* (+ (* (/ 1 (+ x 1)) (/ 1 (+ x 1))) (+ (* (/ 2 x) (/ 2 x)) (* (/ 1 (+ x 1)) (/ 2 x)))) (- x 1))
230.0ms
(* (* (+ x 1) x) (- x 1))

prune407.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize5.0ms

Local error

Found 2 expressions with local error:

3.2b
(+ (* 1 (- (/ 1 (+ x 1)) (/ 2 x))) (/ 1 (- x 1)))
0.0b
(- (/ 1 (+ x 1)) (/ 2 x))

rewrite21.0ms

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

series67.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
57.0ms
(+ (* 1 (- (/ 1 (+ x 1)) (/ 2 x))) (/ 1 (- x 1)))
11.0ms
(- (/ 1 (+ x 1)) (/ 2 x))

simplify2.9s

Counts
26 → 42
Calls
26 calls:
Slowest
542.0ms
(* (+ (/ 1 (+ x 1)) (/ 2 x)) (- x 1))
436.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))
306.0ms
(+ (* (* 1 (- (* 1 x) (* (+ x 1) 2))) (- x 1)) (* (* (+ x 1) x) 1))
258.0ms
(+ (* (* 1 (- (* (/ 1 (+ x 1)) (/ 1 (+ x 1))) (* (/ 2 x) (/ 2 x)))) (- x 1)) (* (+ (/ 1 (+ x 1)) (/ 2 x)) 1))
212.0ms
(* (* (+ x 1) x) (- x 1))

prune356.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

localize37.0ms

Local error

Found 4 expressions with local error:

14.7b
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))
0.1b
(* (* (+ x 1) x) (- x 1))
0.0b
(* (- (* 1 x) (* (+ x 1) 2)) (- x 1))
0.0b
(/ (+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1)) (* (* (+ x 1) x) (- x 1)))

rewrite104.0ms

Algorithm
rewrite-expression-head
Rules
90×associate-*l/
36×flip--
36×flip3--
32×frac-add
28×frac-times
24×flip-+
24×flip3-+
18×associate-/l/
14×associate-*r/
13×add-exp-log
13×add-cbrt-cube
9×add-cube-cbrt
9×*-un-lft-identity
9×add-sqr-sqrt
8×associate-/r/
8×pow1
6×add-log-exp
6×associate-*r*
6×distribute-lft-in
6×sub-neg
6×distribute-rgt-in
4×associate-*l*
4×associate-+l+
3×associate-/l*
3×cbrt-unprod
3×prod-exp
3×times-frac
2×*-commutative
2×pow-prod-down
2×div-exp
2×cbrt-undiv
1×div-inv
1×frac-2neg
1×sum-log
1×associate-/r*
1×clear-num
1×+-commutative
Counts
4 → 134
Calls
4 calls:
Slowest
56.0ms
(/ (+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1)) (* (* (+ x 1) x) (- x 1)))
16.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))
14.0ms
(* (* (+ x 1) x) (- x 1))
12.0ms
(* (- (* 1 x) (* (+ x 1) 2)) (- x 1))

series126.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
63.0ms
(* (* (+ x 1) x) (- x 1))
41.0ms
(/ (+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1)) (* (* (+ x 1) x) (- x 1)))
18.0ms
(* (- (* 1 x) (* (+ x 1) 2)) (- x 1))
3.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))

simplify26.0s

Counts
135 → 146
Calls
135 calls:
Slowest
911.0ms
(+ (* (* (- (pow (* 1 x) 3) (pow (* (+ x 1) 2) 3)) (- (pow x 3) (pow 1 3))) (+ (* x x) (- (* 1 1) (* x 1)))) (* (* (+ (* (* 1 x) (* 1 x)) (+ (* (* (+ x 1) 2) (* (+ x 1) 2)) (* (* 1 x) (* (+ x 1) 2)))) (+ (* x x) (+ (* 1 1) (* x 1)))) (* (* (+ (pow x 3) (pow 1 3)) x) 1)))
723.0ms
(+ (* (* (- (pow (* 1 x) 3) (pow (* (+ x 1) 2) 3)) (- (pow x 3) (pow 1 3))) (+ (* x x) (- (* 1 1) (* x 1)))) (* (* (+ (* (* 1 x) (* 1 x)) (+ (* (* (+ x 1) 2) (* (+ x 1) 2)) (* (* 1 x) (* (+ x 1) 2)))) (+ (* x x) (+ (* 1 1) (* x 1)))) (* (* (+ (pow x 3) (pow 1 3)) x) 1)))
687.0ms
(+ (* (* (- (pow (* 1 x) 3) (pow (* (+ x 1) 2) 3)) (- (pow x 3) (pow 1 3))) (- x 1)) (* (* (+ (* (* 1 x) (* 1 x)) (+ (* (* (+ x 1) 2) (* (+ x 1) 2)) (* (* 1 x) (* (+ x 1) 2)))) (+ (* x x) (+ (* 1 1) (* x 1)))) (* (* (- (* x x) (* 1 1)) x) 1)))
678.0ms
(* (+ (* 1 x) (* (+ x 1) 2)) (+ (* x x) (+ (* 1 1) (* x 1))))
606.0ms
(* (* (- (* x x) (* 1 1)) x) (- x 1))

prune1.2s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

localize3.0ms

Local error

Found 2 expressions with local error:

0.3b
(/ 2 (* (* (+ x 1) x) (- x 1)))
0.1b
(* (* (+ x 1) x) (- x 1))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
16×associate-*l/
8×associate-/r/
8×frac-times
6×flip--
6×flip-+
6×flip3--
6×flip3-+
4×add-exp-log
4×associate-*r/
4×add-cbrt-cube
4×pow1
3×add-cube-cbrt
3×associate-*r*
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
2×distribute-lft-in
2×sub-neg
2×distribute-rgt-in
1×div-inv
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
2 → 46
Calls
2 calls:
Slowest
8.0ms
(* (* (+ x 1) x) (- x 1))
3.0ms
(/ 2 (* (* (+ x 1) x) (- x 1)))

series58.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
36.0ms
(* (* (+ x 1) x) (- x 1))
22.0ms
(/ 2 (* (* (+ x 1) x) (- x 1)))

simplify3.0s

Counts
38 → 52
Calls
38 calls:
Slowest
316.0ms
(* (* (- (* x x) (* 1 1)) x) (- (pow x 3) (pow 1 3)))
264.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))
226.0ms
(* (* (- (* x x) (* 1 1)) x) (- x 1))
206.0ms
(* (* (+ (pow x 3) (pow 1 3)) x) (- (* x x) (* 1 1)))
181.0ms
(* (* (+ (pow x 3) (pow 1 3)) x) (- (pow x 3) (pow 1 3)))

prune259.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

regimes21.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms

end0.0ms

sample1.7s

Algorithm
intervals
Results
560.0ms1848×body640valid
346.0ms4196×body80valid
219.0ms941×body320valid
211.0ms531×body1280valid
61.0ms488×body160valid