Average Error: 10.1 → 0.1
Time: 39.3s
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

Original10.1
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.1

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  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 0.2

    \[\leadsto \frac{\color{blue}{2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
  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 2019022 
(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: 39.0s)Debug log

sample73.0ms

Algorithm
intervals
Results
27.0ms66×body640valid
14.0ms133×body80valid
11.0ms35×body320valid
6.0ms12×body1280valid
2.0ms11×body160valid

simplify49.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
49.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))

prune25.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 12.0b

localize19.0ms

Local error

Found 4 expressions with local error:

2.2b
(+ (- (/ 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))

rewrite40.0ms

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
32.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
5.0ms
(- (/ 1 (+ x 1)) (/ 2 x))
1.0ms
(/ 1 (+ x 1))
1.0ms
(/ 1 (- x 1))

series78.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
51.0ms
(+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))
11.0ms
(- (/ 1 (+ x 1)) (/ 2 x))
9.0ms
(/ 1 (+ x 1))
7.0ms
(/ 1 (- x 1))

simplify3.1s

Counts
48 → 88
Calls
48 calls:
Slowest
509.0ms
(* (+ (/ 1 (+ x 1)) (/ 2 x)) (- x 1))
347.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))
301.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))
238.0ms
(* (* (+ x 1) x) (- x 1))
220.0ms
(* (+ (* (/ 1 (+ x 1)) (/ 1 (+ x 1))) (+ (* (/ 2 x) (/ 2 x)) (* (/ 1 (+ x 1)) (/ 2 x)))) (- x 1))

prune637.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize26.0ms

Local error

Found 4 expressions with local error:

15.6b
(+ (* (- (* 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)) (* (* (+ x 1) x) 1)) (* (* (+ x 1) x) (- x 1)))
0.0b
(* (+ x 1) x)

rewrite62.0ms

Algorithm
rewrite-expression-head
Rules
90×associate-*l/
32×frac-add
30×flip--
30×flip3--
25×flip-+
25×flip3-+
24×frac-times
18×associate-/l/
12×associate-*r/
11×add-exp-log
11×add-cbrt-cube
8×add-cube-cbrt
8×associate-/r/
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-log-exp
6×pow1
4×associate-*l*
4×distribute-lft-in
4×sub-neg
4×associate-+l+
4×distribute-rgt-in
3×associate-/l*
3×associate-*r*
3×times-frac
2×cbrt-unprod
2×*-commutative
2×prod-exp
2×div-exp
2×cbrt-undiv
1×div-inv
1×pow-prod-down
1×frac-2neg
1×sum-log
1×associate-/r*
1×clear-num
1×+-commutative
Counts
4 → 118
Calls
4 calls:
Slowest
38.0ms
(/ (+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1)) (* (* (+ x 1) x) (- x 1)))
8.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))
7.0ms
(* (* (+ x 1) x) (- x 1))
4.0ms
(* (+ x 1) x)

series89.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
39.0ms
(* (* (+ x 1) x) (- x 1))
25.0ms
(* (+ x 1) x)
23.0ms
(/ (+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1)) (* (* (+ x 1) x) (- x 1)))
3.0ms
(+ (* (- (* 1 x) (* (+ x 1) 2)) (- x 1)) (* (* (+ x 1) x) 1))

simplify23.8s

Counts
115 → 130
Calls
115 calls:
Slowest
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)))
960.0ms
(+ (* (* (- (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)))
839.0ms
(+ (* (* (- (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)))
736.0ms
(+ (* (* (- (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)))
648.0ms
(+ (* (* (- (* (* 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)))

prune1.3s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize3.0ms

Local error

Found 3 expressions with local error:

0.2b
(/ 2 (* (* (+ x 1) x) (- x 1)))
0.1b
(* (* (+ x 1) x) (- x 1))
0.0b
(* (+ x 1) x)

rewrite18.0ms

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
10.0ms
(* (* (+ x 1) x) (- x 1))
5.0ms
(/ 2 (* (* (+ x 1) x) (- x 1)))
2.0ms
(* (+ x 1) x)

series97.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
44.0ms
(* (* (+ x 1) x) (- x 1))
27.0ms
(/ 2 (* (* (+ x 1) x) (- x 1)))
25.0ms
(* (+ x 1) x)

simplify3.7s

Counts
45 → 68
Calls
45 calls:
Slowest
388.0ms
(* (* (- (* x x) (* 1 1)) x) (- (pow x 3) (pow 1 3)))
313.0ms
(* (* (- (* x x) (* 1 1)) x) (- x 1))
280.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))
273.0ms
(* (* (+ (pow x 3) (pow 1 3)) x) (- (* x x) (* 1 1)))
225.0ms
(* (* (+ x 1) x) (- x 1))

prune428.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 3 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
18×times-frac
15×add-cube-cbrt
15×*-un-lft-identity
15×add-sqr-sqrt
12×associate-/r/
12×associate-*l/
6×associate-/l*
6×div-inv
6×flip-+
6×flip3-+
5×add-exp-log
5×add-cbrt-cube
4×associate-/r*
3×add-log-exp
3×associate-*l*
3×pow1
2×frac-2neg
2×clear-num
1×associate-/l/
1×flip--
1×*-commutative
1×div-exp
1×flip3--
1×cbrt-undiv
Counts
3 → 68
Calls
3 calls:
Slowest
8.0ms
(/ (/ 2 (* (+ x 1) x)) (- x 1))
4.0ms
(* (+ x 1) x)
1.0ms
(/ 2 (* (+ x 1) x))

series59.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
26.0ms
(/ (/ 2 (* (+ x 1) x)) (- x 1))
19.0ms
(* (+ x 1) x)
13.0ms
(/ 2 (* (+ x 1) x))

simplify3.4s

Counts
62 → 77
Calls
62 calls:
Slowest
475.0ms
(- (+ (* 2 (/ 1 (pow x 4))) (* 2 (/ 1 (pow x 2)))) (* 2 (/ 1 (pow x 3))))
249.0ms
(- (+ (* 2 x) (+ (* 2 (pow x 3)) (* 2 (/ 1 x)))))
224.0ms
(- (+ (* 2 (/ 1 (pow x 4))) (* 2 (/ 1 (pow x 2)))) (* 2 (/ 1 (pow x 3))))
184.0ms
(/ (/ 2 (* (+ (pow x 3) (pow 1 3)) x)) (sqrt (- x 1)))
179.0ms
(/ (/ 2 (* (- (* x x) (* 1 1)) x)) (* (cbrt (- x 1)) (cbrt (- x 1))))

prune359.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

regimes25.0ms

Accuracy

0% (0.1b remaining)

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

bsearch1.0ms

end0.0ms

sample1.6s

Algorithm
intervals
Results
656.0ms1856×body640valid
333.0ms4179×body80valid
233.0ms953×body320valid
200.0ms540×body1280valid
64.0ms477×body160valid