Average Error: 9.5 → 0.1
Time: 32.4s
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

Original9.5
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.5

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  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-add25.4

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

  5. Taylor expanded around -inf 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 2019007 
(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: 31.9s)Debug log

sample101.0ms

Algorithm
intervals

simplify61.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

61.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 8.7b

localize48.0ms

Local error

Found 4 expressions with local error:

2.4b
(+ (- (/ 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))

rewrite44.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

37.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
5.0ms
(- (/ 1 (+ x 1)) (/ 2 x))
1.0ms
(/ 1 (+ x 1))

series62.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

37.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
11.0ms
(- (/ 1 (+ x 1)) (/ 2 x))
8.0ms
(/ 1 (+ x 1))
6.0ms
(/ 1 (- x 1))

simplify3.7s

Counts
48 → 88
Calls

48 calls. Slowest were:

719.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))
424.0ms
(* (+ (/ 1 (+ x 1)) (/ 2 x)) (- x 1))
417.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))

prune770.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize27.0ms

Local error

Found 4 expressions with local error:

17.1b
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ (/ 1 (* (cbrt (- x 1)) (cbrt (- x 1)))) (cbrt (- x 1))))
0.3b
(cbrt (- x 1))
0.3b
(cbrt (- x 1))
0.3b
(cbrt (- x 1))

rewrite42.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

40.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ (/ 1 (* (cbrt (- x 1)) (cbrt (- x 1)))) (cbrt (- x 1))))
1.0ms
(cbrt (- x 1))
1.0ms
(cbrt (- x 1))

series225.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

92.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ (/ 1 (* (cbrt (- x 1)) (cbrt (- x 1)))) (cbrt (- x 1))))
49.0ms
(cbrt (- x 1))
45.0ms
(cbrt (- x 1))
39.0ms
(cbrt (- x 1))

simplify3.8s

Counts
41 → 72
Calls

41 calls. Slowest were:

384.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))
371.0ms
(- (pow -1 1/3) (+ (* 1/3 (* x (pow -1 1/3))) (* 1/9 (* (pow x 2) (pow -1 1/3)))))
355.0ms
(- (pow -1 1/3) (+ (* 1/3 (* x (pow -1 1/3))) (* 1/9 (* (pow x 2) (pow -1 1/3)))))

prune740.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.1b

localize43.0ms

Local error

Found 4 expressions with local error:

18.4b
(+ (* (- (* 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))
0.0b
(* (+ x 1) x)

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

8.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))
7.0ms
(* (* (+ x 1) x) (- x 1))
2.0ms
(* (+ x 1) x)

series91.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

59.0ms
(* (* (+ x 1) x) (- x 1))
18.0ms
(* (+ x 1) x)
11.0ms
(- (* 1 x) (* (+ x 1) 2))
2.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))

simplify13.6s

Counts
77 → 95
Calls

77 calls. Slowest were:

1.0s
(+ (* (* (- (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)))
800.0ms
(+ (* (* (- (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)))
642.0ms
(+ (* (* (- (* (* 1 x) (* 1 x)) (* (* (+ x 1) 2) (* (+ x 1) 2))) (- (pow x 3) (pow 1 3))) (+ (* x x) (- (* 1 1) (* x 1)))) (* (* (+ (* 1 x) (* (+ x 1) 2)) (+ (* x x) (+ (* 1 1) (* x 1)))) (* (* (+ (pow x 3) (pow 1 3)) x) 1)))

prune1.0s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize3.0ms

Local error

Found 3 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
3 → 59
Calls

3 calls. Slowest were:

7.0ms
(* (* (+ x 1) x) (- x 1))
3.0ms
(/ 2 (* (* (+ x 1) x) (- x 1)))
2.0ms
(* (+ x 1) x)

series88.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

37.0ms
(* (* (+ x 1) x) (- x 1))
26.0ms
(/ 2 (* (* (+ x 1) x) (- x 1)))
25.0ms
(* (+ x 1) x)

simplify4.9s

Counts
45 → 68
Calls

45 calls. Slowest were:

537.0ms
(* (* (- (* x x) (* 1 1)) x) (- (* x x) (* 1 1)))
518.0ms
(* (* (- (* x x) (* 1 1)) x) (- (pow x 3) (pow 1 3)))
430.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))

prune524.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

regimes22.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms

end0.0ms

sample2.0s

Algorithm
intervals