Average Error: 14.2 → 0.4
Time: 15.1s
Precision: 64
Internal Precision: 128
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{-1}{(x \cdot x + x)_*}\]

Error

Bits error versus x

Derivation

  1. Initial program 14.2

    \[\frac{1}{x + 1} - \frac{1}{x}\]

  2. Initial simplification14.2

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

  4. Applied frac-sub13.6

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

  5. Simplified0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

    \[\leadsto \frac{-1}{(x \cdot x + x)_*}\]

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Details

Time bar (total: 10.6s)Debug log

start54.0ms

Algorithm
intervals

setup44.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 10.6b

localize24.0ms

Local error

Found 2 expressions with local error:

1.6b
(- (/ 1 (+ x 1)) (/ 1 x))
0.0b
(/ 1 (+ x 1))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
2 → 70
Calls

2 calls. Slowest were:

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

series28.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

18.0ms
(- (/ 1 (+ x 1)) (/ 1 x))
10.0ms
(/ 1 (+ x 1))

simplify3.8s

Counts
73 → 76
Calls

73 calls. Slowest were:

539.0ms
(fma (/ 1 (+ (pow x 3) (pow 1 3))) (+ (* x x) (- (* 1 1) (* x 1))) (- (* (/ 1 x) 1)))
538.0ms
(fma (/ 1 (+ (pow x 3) (pow 1 3))) (+ (* x x) (- (* 1 1) (* x 1))) (- (* (sqrt (/ 1 x)) (sqrt (/ 1 x)))))
440.0ms
(fma (/ 1 (+ (pow x 3) (pow 1 3))) (+ (* x x) (- (* 1 1) (* x 1))) (- (* (/ 1 x) 1)))

prune549.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.1b

localize13.0ms

Local error

Found 1 expressions with local error:

0.1b
(/ -1 (fma x x x))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 15
Calls

1 calls. Slowest were:

1.0ms
(/ -1 (fma x x x))

series19.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

19.0ms
(/ -1 (fma x x x))

simplify193.0ms

Counts
6 → 18
Calls

6 calls. Slowest were:

76.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
60.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
48.0ms
(- 1 (+ x (/ 1 x)))

prune187.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.1b

localize18.0ms

Local error

Found 4 expressions with local error:

13.0b
(sqrt (fma x x x))
13.0b
(sqrt (fma x x x))
0.2b
(/ (/ -1 (sqrt (fma x x x))) (sqrt (fma x x x)))
0.1b
(/ -1 (sqrt (fma x x x)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 95
Calls

4 calls. Slowest were:

8.0ms
(/ (/ -1 (sqrt (fma x x x))) (sqrt (fma x x x)))
2.0ms
(sqrt (fma x x x))
2.0ms
(sqrt (fma x x x))

series105.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

34.0ms
(/ -1 (sqrt (fma x x x)))
25.0ms
(sqrt (fma x x x))
25.0ms
(sqrt (fma x x x))
20.0ms
(/ (/ -1 (sqrt (fma x x x))) (sqrt (fma x x x)))

simplify2.4s

Counts
84 → 107
Calls

84 calls. Slowest were:

280.0ms
(/ (* (* (/ -1 (sqrt (fma x x x))) (/ -1 (sqrt (fma x x x)))) (/ -1 (sqrt (fma x x x)))) (* (* (sqrt (fma x x x)) (sqrt (fma x x x))) (sqrt (fma x x x))))
225.0ms
(- (+ (* 3/8 (/ 1 (pow x 3))) (/ 1 x)) (* 1/2 (/ 1 (pow x 2))))
197.0ms
(- (* 1/2 (/ 1 (pow x 2))) (+ (* 3/8 (/ 1 (pow x 3))) (/ 1 x)))

prune970.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.1b

localize11.0ms

Local error

Found 4 expressions with local error:

13.0b
(sqrt (fma x x x))
13.0b
(sqrt (fma x x x))
13.0b
(sqrt (fma x x x))
0.3b
(* (sqrt (sqrt (fma x x x))) (sqrt (sqrt (fma x x x))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 80
Calls

4 calls. Slowest were:

15.0ms
(* (sqrt (sqrt (fma x x x))) (sqrt (sqrt (fma x x x))))
2.0ms
(sqrt (fma x x x))
2.0ms
(sqrt (fma x x x))

series82.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

26.0ms
(* (sqrt (sqrt (fma x x x))) (sqrt (sqrt (fma x x x))))
24.0ms
(sqrt (fma x x x))
17.0ms
(sqrt (fma x x x))
15.0ms
(sqrt (fma x x x))

simplify1.0s

Counts
52 → 92
Calls

52 calls. Slowest were:

101.0ms
(- (+ x 1/2) (* 1/8 (/ 1 x)))
96.0ms
(- (+ x 1/2) (* 1/8 (/ 1 x)))
84.0ms
(- (+ x 1/2) (* 1/8 (/ 1 x)))

prune1.0s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.1b

regimes62.0ms

Accuracy

0% (0.1b remaining)

Error of 0.4b against oracle of 0.3b and baseline of 0.4b

bsearch1.0ms