Average Error: 9.9 → 0.1
Time: 1.2m
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
Target0.3
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-sub26.1

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

  4. Applied frac-add25.6

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

  5. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
  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 323378853 
(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: 1.1m)Debug log

sample122.0ms

Algorithm
intervals
Results
48.0ms65×body640valid
25.0ms134×body80valid
17.0ms18×body1280valid
15.0ms27×body320valid
4.0ms12×body160valid

simplify81.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
81.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 10.5b

localize52.0ms

Local error

Found 4 expressions with local error:

1.8b
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
0.0b
(- (/ 1 (+ x 1)) (/ 2 x))
0.0b
(/ 1 (+ x 1))
0.0b
(/ 1 (- x 1))

rewrite71.0ms

Algorithm
rewrite-expression-head
Rules
20×*-un-lft-identity
11×add-log-exp
8×div-inv
8×add-sqr-sqrt
6×distribute-lft-out--
6×distribute-lft-out
6×add-cube-cbrt
6×add-exp-log
6×associate-/r*
6×pow1
4×associate-/r/
4×add-cbrt-cube
3×flip--
3×frac-add
3×flip3--
2×inv-pow
2×pow-flip
2×flip-+
2×frac-sub
2×diff-log
2×flip3-+
2×frac-2neg
2×sub-neg
2×sum-log
2×clear-num
2×associate-+l+
2×rec-exp
1×difference-of-squares
1×associate-+l-
1×+-commutative
Counts
4 → 76
Calls
4 calls:
Slowest
54.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
11.0ms
(- (/ 1 (+ x 1)) (/ 2 x))
2.0ms
(/ 1 (+ x 1))
2.0ms
(/ 1 (- x 1))

series120.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
73.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
22.0ms
(- (/ 1 (+ x 1)) (/ 2 x))
13.0ms
(/ 1 (- x 1))
13.0ms
(/ 1 (+ x 1))

simplify4.7s

Counts
48 → 88
Calls
48 calls:
Slowest
613.0ms
(* (+ (/ 1 (+ x 1)) (/ 2 x)) (- x 1))
555.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))
464.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))
371.0ms
(* (* (+ x 1) x) (- x 1))
350.0ms
(* (+ (* (/ 1 (+ x 1)) (/ 1 (+ x 1))) (+ (* (/ 2 x) (/ 2 x)) (* (/ 1 (+ x 1)) (/ 2 x)))) (- x 1))

prune1.1s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize24.0ms

Local error

Found 4 expressions with local error:

1.8b
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ (/ 1 1) (- x 1)))
0.0b
(- (/ 1 (+ x 1)) (/ 2 x))
0.0b
(/ 1 (+ x 1))
0.0b
(/ (/ 1 1) (- x 1))

rewrite75.0ms

Algorithm
rewrite-expression-head
Rules
24×*-un-lft-identity
16×add-sqr-sqrt
14×add-cube-cbrt
12×times-frac
11×add-log-exp
8×div-inv
7×add-exp-log
6×add-cbrt-cube
6×associate-/r*
5×pow1
4×distribute-lft-out--
4×associate-/l*
4×associate-/r/
3×distribute-lft-out
3×flip--
3×frac-add
3×flip3--
2×flip-+
2×frac-sub
2×diff-log
2×flip3-+
2×frac-2neg
2×sub-neg
2×sum-log
2×clear-num
2×associate-+l+
1×associate-/l/
1×difference-of-squares
1×inv-pow
1×pow-flip
1×associate-+l-
1×div-exp
1×rec-exp
1×+-commutative
1×cbrt-undiv
Counts
4 → 89
Calls
4 calls:
Slowest
52.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ (/ 1 1) (- x 1)))
11.0ms
(- (/ 1 (+ x 1)) (/ 2 x))
7.0ms
(/ (/ 1 1) (- x 1))
2.0ms
(/ 1 (+ x 1))

series125.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
77.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ (/ 1 1) (- x 1)))
21.0ms
(- (/ 1 (+ x 1)) (/ 2 x))
14.0ms
(/ (/ 1 1) (- x 1))
13.0ms
(/ 1 (+ x 1))

simplify5.4s

Counts
72 → 101
Calls
72 calls:
Slowest
604.0ms
(* (+ (/ 1 (+ x 1)) (/ 2 x)) (- x 1))
553.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))
476.0ms
(+ (* (- (pow (/ 1 (+ x 1)) 3) (pow (/ 2 x) 3)) (- x 1)) (* (+ (* (/ 1 (+ x 1)) (/ 1 (+ x 1))) (+ (* (/ 2 x) (/ 2 x)) (* (/ 1 (+ x 1)) (/ 2 x)))) (/ 1 1)))
418.0ms
(/ (* (* (/ 1 1) (/ 1 1)) (/ 1 1)) (* (* (- x 1) (- x 1)) (- x 1)))
373.0ms
(* (* (+ x 1) x) (- x 1))

prune1.3s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

localize64.0ms

Local error

Found 4 expressions with local error:

15.2b
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))
0.0b
(* (* (+ 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)))

rewrite127.0ms

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
62.0ms
(/ (+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1)) (* (* (+ x 1) x) (- x 1)))
25.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))
18.0ms
(* (* (+ x 1) x) (- x 1))
15.0ms
(* (- (* 1 x) (* (+ x 1) 2)) (- x 1))

series150.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
77.0ms
(* (* (+ x 1) x) (- x 1))
48.0ms
(/ (+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1)) (* (* (+ x 1) x) (- x 1)))
20.0ms
(* (- (* 1 x) (* (+ x 1) 2)) (- x 1))
4.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))

simplify40.7s

Counts
135 → 146
Calls
135 calls:
Slowest
1.3s
(+ (* (* (- (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)))
1.3s
(+ (* (* (- (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)))
1.0s
(+ (* (* (- (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)))
1.0s
(+ (* (* (- (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)))
908.0ms
(+ (pow (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) 3) (pow (* (* (+ x 1) x) 1) 3))

prune2.5s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 3 expressions with local error:

0.3b
(/ 2 (* (* (+ x 1) x) (- x 1)))
0.0b
(* (* (+ x 1) x) (- x 1))
0.0b
(* (+ x 1) x)

rewrite32.0ms

Algorithm
rewrite-expression-head
Rules
18×associate-*l/
8×associate-/r/
8×frac-times
7×flip-+
7×flip3-+
6×flip--
6×flip3--
5×add-cube-cbrt
5×add-exp-log
5×add-cbrt-cube
5×*-un-lft-identity
5×pow1
5×add-sqr-sqrt
4×associate-*r/
4×associate-*l*
3×add-log-exp
3×associate-*r*
2×*-commutative
2×distribute-lft-in
2×sub-neg
2×distribute-rgt-in
1×div-inv
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
3 → 59
Calls
3 calls:
Slowest
18.0ms
(* (* (+ x 1) x) (- x 1))
6.0ms
(/ 2 (* (* (+ x 1) x) (- x 1)))
6.0ms
(* (+ x 1) x)

series160.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
73.0ms
(* (* (+ x 1) x) (- x 1))
49.0ms
(/ 2 (* (* (+ x 1) x) (- x 1)))
37.0ms
(* (+ x 1) x)

simplify6.4s

Counts
45 → 68
Calls
45 calls:
Slowest
644.0ms
(* (* (- (* x x) (* 1 1)) x) (- (pow x 3) (pow 1 3)))
557.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))
465.0ms
(* (* (- (* x x) (* 1 1)) x) (- x 1))
429.0ms
(* (* (+ (pow x 3) (pow 1 3)) x) (- (* x x) (* 1 1)))
383.0ms
(* (* (+ (pow x 3) (pow 1 3)) x) (- (pow x 3) (pow 1 3)))

prune795.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

regimes51.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms

end0.0ms

sample4.2s

Algorithm
intervals
Results
1.6s1836×body640valid
1.2s4091×body80valid
599.0ms644×body1280valid
560.0ms969×body320valid
195.0ms460×body160valid