Average Error: 14.8 → 0.4
Time: 29.4s
Precision: 64
Internal Precision: 128
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{1}{(x \cdot x + -1)_*} \cdot -2\]

Error

Bits error versus x

Derivation

  1. Initial program 14.8

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

  3. Applied flip--29.1

    \[\leadsto \frac{1}{x + 1} - \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}}\]

  4. Applied associate-/r/29.2

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

  5. Applied flip-+14.8

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

  6. Applied associate-/r/14.8

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

  7. Applied distribute-lft-out--14.2

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

  8. Simplified14.2

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

  9. Simplified0.4

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

  10. Final simplification0.4

    \[\leadsto \frac{1}{(x \cdot x + -1)_*} \cdot -2\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))

Details

Time bar (total: 29.0s)Debug log

sample56.0ms

Algorithm
intervals

simplify13.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

13.0ms
(- (/ 1 (+ x 1)) (/ 1 (- x 1)))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 14.7b

localize23.0ms

Local error

Found 3 expressions with local error:

2.6b
(- (/ 1 (+ x 1)) (/ 1 (- x 1)))
0.0b
(/ 1 (+ x 1))
0.0b
(/ 1 (- x 1))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 103
Calls

3 calls. Slowest were:

7.0ms
(- (/ 1 (+ x 1)) (/ 1 (- x 1)))
1.0ms
(/ 1 (+ x 1))
1.0ms
(/ 1 (- x 1))

series56.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

26.0ms
(- (/ 1 (+ x 1)) (/ 1 (- x 1)))
20.0ms
(/ 1 (- x 1))
10.0ms
(/ 1 (+ x 1))

simplify19.0s

Counts
108 → 112
Calls

108 calls. Slowest were:

1.2s
(fma (- (+ x 1)) (/ 1 (- (* x x) (* 1 1))) (* (+ x 1) (/ 1 (- (* x x) (* 1 1)))))
1.1s
(fma (- (+ x 1)) (/ 1 (- (* x x) (* 1 1))) (* (+ x 1) (/ 1 (- (* x x) (* 1 1)))))
1.0s
(fma (- (+ x 1)) (/ 1 (- (* x x) (* 1 1))) (* (+ x 1) (/ 1 (- (* x x) (* 1 1)))))

prune766.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize21.0ms

Local error

Found 2 expressions with local error:

0.2b
(/ 1 (fma x x -1))
0.0b
(* (/ 1 (fma x x -1)) -2)

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 33
Calls

2 calls. Slowest were:

6.0ms
(* (/ 1 (fma x x -1)) -2)
2.0ms
(/ 1 (fma x x -1))

series47.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

24.0ms
(* (/ 1 (fma x x -1)) -2)
23.0ms
(/ 1 (fma x x -1))

simplify2.0s

Counts
13 → 39
Calls

13 calls. Slowest were:

838.0ms
(- (+ (* 2 (/ 1 (pow x 4))) (+ (* 2 (/ 1 (pow x 6))) (* 2 (/ 1 (pow x 2))))))
823.0ms
(- (+ (* 2 (/ 1 (pow x 4))) (+ (* 2 (/ 1 (pow x 6))) (* 2 (/ 1 (pow x 2))))))
103.0ms
(+ (* 2 (pow x 2)) (+ (* 2 (pow x 4)) 2))

prune299.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.1b

localize11.0ms

Local error

Found 4 expressions with local error:

14.5b
(cbrt (/ 1 (fma x x -1)))
14.5b
(cbrt (/ 1 (fma x x -1)))
14.5b
(cbrt (/ 1 (fma x x -1)))
0.2b
(/ 1 (fma x x -1))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

3.0ms
(cbrt (/ 1 (fma x x -1)))
1.0ms
(/ 1 (fma x x -1))
1.0ms
(cbrt (/ 1 (fma x x -1)))

series323.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

120.0ms
(cbrt (/ 1 (fma x x -1)))
101.0ms
(cbrt (/ 1 (fma x x -1)))
87.0ms
(cbrt (/ 1 (fma x x -1)))
15.0ms
(/ 1 (fma x x -1))

simplify1.2s

Counts
31 → 75
Calls

31 calls. Slowest were:

204.0ms
(+ (* 1/3 (* (pow x 2) (pow -1 1/3))) (+ (* 2/9 (* (pow x 4) (pow -1 1/3))) (pow -1 1/3)))
184.0ms
(+ (* 1/3 (* (pow x 2) (pow -1 1/3))) (+ (* 2/9 (* (pow x 4) (pow -1 1/3))) (pow -1 1/3)))
163.0ms
(+ (* 1/3 (* (pow x 2) (pow -1 1/3))) (+ (* 2/9 (* (pow x 4) (pow -1 1/3))) (pow -1 1/3)))

prune715.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.1b

localize14.0ms

Local error

Found 4 expressions with local error:

14.6b
(cbrt (fma x x -1))
14.5b
(cbrt (/ 1 (fma x x -1)))
14.5b
(cbrt (/ 1 (fma x x -1)))
0.2b
(/ 1 (fma x x -1))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

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

series414.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

152.0ms
(cbrt (/ 1 (fma x x -1)))
126.0ms
(cbrt (/ 1 (fma x x -1)))
111.0ms
(cbrt (fma x x -1))
24.0ms
(/ 1 (fma x x -1))

simplify1.4s

Counts
29 → 73
Calls

29 calls. Slowest were:

258.0ms
(- (pow -1 1/3) (+ (* 1/3 (* (pow x 2) (pow -1 1/3))) (* 1/9 (* (pow x 4) (pow -1 1/3)))))
191.0ms
(+ (* 1/3 (* (pow x 2) (pow -1 1/3))) (+ (* 2/9 (* (pow x 4) (pow -1 1/3))) (pow -1 1/3)))
177.0ms
(+ (* 1/3 (* (pow x 2) (pow -1 1/3))) (+ (* 2/9 (* (pow x 4) (pow -1 1/3))) (pow -1 1/3)))

prune812.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.1b

regimes45.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms

end0.0ms

sample1.9s

Algorithm
intervals