Average Error: 15.1 → 0.4
Time: 7.5s
Precision: 64
Internal Precision: 128
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{(N \cdot \left(N + 1\right) + 1)_*}\]

Error

Bits error versus N

Target

Original15.1
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.1

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
  2. Using strategy rm

  3. Applied diff-atan14.0

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

    \[\leadsto \tan^{-1}_* \frac{1}{(N \cdot \left(N + 1\right) + 1)_*}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (N)
  :name "2atan (example 3.5)"

  :herbie-target
  (atan (/ 1 (+ 1 (* N (+ N 1)))))

  (- (atan (+ N 1)) (atan N)))

Details

Time bar (total: 6.7s)Debug log

sample76.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

8.0ms
(- (atan (+ N 1)) (atan N))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 15.5b

localize22.0ms

Local error

Found 2 expressions with local error:

1.6b
(- (atan (+ N 1)) (atan N))
0.0b
(atan (+ N 1))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
2 → 38
Calls

2 calls. Slowest were:

8.0ms
(- (atan (+ N 1)) (atan N))
1.0ms
(atan (+ N 1))

series14.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

11.0ms
(- (atan (+ N 1)) (atan N))
3.0ms
(atan (+ N 1))

simplify594.0ms

Counts
32 → 44
Calls

32 calls. Slowest were:

48.0ms
(fma (- (cbrt (atan N))) (* (cbrt (atan N)) (cbrt (atan N))) (* (cbrt (atan N)) (* (cbrt (atan N)) (cbrt (atan N)))))
42.0ms
(fma (sqrt (atan (+ N 1))) (sqrt (atan (+ N 1))) (- (* (sqrt (atan N)) (sqrt (atan N)))))
34.0ms
(fma 1 (atan (+ N 1)) (- (* (atan N) 1)))

prune379.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize11.0ms

Local error

Found 1 expressions with local error:

0.5b
(atan2 1 (fma N (+ N 1) 1))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 9
Calls

1 calls. Slowest were:

0.0ms
(atan2 1 (fma N (+ N 1) 1))

series2.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

2.0ms
(atan2 1 (fma N (+ N 1) 1))

simplify35.0ms

Counts
3 → 12
Calls

3 calls. Slowest were:

13.0ms
(atan2 1 (fma N (+ N 1) 1))
11.0ms
(atan2 1 (fma N (+ N 1) 1))
11.0ms
(atan2 1 (fma N (+ N 1) 1))

prune108.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.4b

localize5.0ms

Local error

Found 4 expressions with local error:

17.6b
(sqrt (atan2 1 (fma N (+ N 1) 1)))
17.6b
(sqrt (atan2 1 (fma N (+ N 1) 1)))
0.5b
(* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1))))
0.5b
(atan2 1 (fma N (+ N 1) 1))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

5.0ms
(* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1))))
1.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))
1.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))

series39.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

20.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))
14.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))
2.0ms
(* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1))))
2.0ms
(atan2 1 (fma N (+ N 1) 1))

simplify859.0ms

Counts
42 → 81
Calls

42 calls. Slowest were:

59.0ms
(* (* (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))) (sqrt (atan2 1 (fma N (+ N 1) 1)))) (* (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
41.0ms
(+ (log (sqrt (atan2 1 (fma N (+ N 1) 1)))) (log (sqrt (atan2 1 (fma N (+ N 1) 1)))))
38.0ms
(* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1))))

prune759.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.4b

localize49.0ms

Local error

Found 4 expressions with local error:

18.1b
(cbrt (atan2 1 (+ 1 (fma N N N))))
18.1b
(cbrt (atan2 1 (fma N (+ N 1) 1)))
17.6b
(sqrt (atan2 1 (fma N (+ N 1) 1)))
0.5b
(atan2 1 (+ 1 (fma N N N)))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 49
Calls

4 calls. Slowest were:

2.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))
1.0ms
(cbrt (atan2 1 (+ 1 (fma N N N))))
1.0ms
(cbrt (atan2 1 (fma N (+ N 1) 1)))

series404.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

193.0ms
(cbrt (atan2 1 (+ 1 (fma N N N))))
189.0ms
(cbrt (atan2 1 (fma N (+ N 1) 1)))
18.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))
4.0ms
(atan2 1 (+ 1 (fma N N N)))

simplify466.0ms

Counts
22 → 61
Calls

22 calls. Slowest were:

40.0ms
(cbrt (* (cbrt (atan2 1 (+ 1 (fma N N N)))) (cbrt (atan2 1 (+ 1 (fma N N N))))))
36.0ms
(cbrt (sqrt (atan2 1 (+ 1 (fma N N N)))))
32.0ms
(pow (atan2 1 (+ (fma N N N) 1)) 1/3)

prune950.0ms

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0.4b

regimes47.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample1.9s

Algorithm
intervals