Average Error: 14.6 → 0.4
Time: 6.4s
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

Original14.6
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.6

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

  3. Applied diff-atan13.6

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

  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 2019007 +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: 6.0s)Debug log

sample68.0ms

Algorithm
intervals

simplify4.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

4.0ms
(- (atan (+ N 1)) (atan N))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 12.3b

localize11.0ms

Local error

Found 2 expressions with local error:

1.0b
(- (atan (+ N 1)) (atan N))
0.0b
(atan (+ N 1))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 38
Calls

2 calls. Slowest were:

4.0ms
(- (atan (+ N 1)) (atan N))
1.0ms
(atan (+ N 1))

series10.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

8.0ms
(- (atan (+ N 1)) (atan N))
2.0ms
(atan (+ N 1))

simplify473.0ms

Counts
32 → 44
Calls

32 calls. Slowest were:

44.0ms
(fma (* (cbrt (atan (+ N 1))) (cbrt (atan (+ N 1)))) (cbrt (atan (+ N 1))) (- (* (cbrt (atan N)) (* (cbrt (atan N)) (cbrt (atan N))))))
30.0ms
(fma (- (cbrt (atan N))) (* (cbrt (atan N)) (cbrt (atan N))) (* (cbrt (atan N)) (* (cbrt (atan N)) (cbrt (atan N)))))
28.0ms
(fma (sqrt (atan (+ N 1))) (sqrt (atan (+ N 1))) (- (* (cbrt (atan N)) (* (cbrt (atan N)) (cbrt (atan N))))))

prune296.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.2b

localize17.0ms

Local error

Found 1 expressions with local error:

0.2b
(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))

series3.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

3.0ms
(atan2 1 (fma N (+ N 1) 1))

simplify49.0ms

Counts
3 → 12
Calls

3 calls. Slowest were:

21.0ms
(atan2 1 (fma N (+ N 1) 1))
16.0ms
(atan2 1 (fma N (+ N 1) 1))
10.0ms
(atan2 1 (fma N (+ N 1) 1))

prune84.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.2b

localize5.0ms

Local error

Found 4 expressions with local error:

17.1b
(sqrt (atan2 1 (fma N (+ N 1) 1)))
17.1b
(sqrt (atan2 1 (fma N (+ N 1) 1)))
0.6b
(* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1))))
0.2b
(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)))

series28.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

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

simplify802.0ms

Counts
42 → 81
Calls

42 calls. Slowest were:

47.0ms
(* (* (* (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)))))
40.0ms
(cbrt (sqrt (atan2 1 (fma N (+ N 1) 1))))
37.0ms
(sqrt (sqrt (atan2 1 (fma N (+ N 1) 1))))

prune617.0ms

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.2b

localize12.0ms

Local error

Found 4 expressions with local error:

17.1b
(sqrt (atan2 1 (fma N (+ N 1) 1)))
17.1b
(sqrt (atan2 1 (fma N (+ N 1) 1)))
17.1b
(sqrt (atan2 1 (fma N (+ N 1) 1)))
0.6b
(* (sqrt (atan2 1 (fma N (+ N 1) 1))) (* (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1)))) (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1))))))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

13.0ms
(* (sqrt (atan2 1 (fma N (+ N 1) 1))) (* (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1)))) (sqrt (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)))

series38.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

14.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))
11.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))
10.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))
3.0ms
(* (sqrt (atan2 1 (fma N (+ N 1) 1))) (* (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1)))) (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1))))))

simplify881.0ms

Counts
40 → 80
Calls

40 calls. Slowest were:

98.0ms
(* (* (* (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 (sqrt (atan2 1 (fma N (+ N 1) 1)))) (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1))))) (* (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1)))) (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1)))))) (* (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1)))) (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1)))))))
84.0ms
(* (* (* (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 (sqrt (atan2 1 (fma N (+ N 1) 1)))) (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1))))) (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1))))) (* (* (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1)))) (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1))))) (sqrt (sqrt (atan2 1 (fma N (+ N 1) 1)))))))
39.0ms
(sqrt (* (cbrt (atan2 1 (fma N (+ N 1) 1))) (cbrt (atan2 1 (fma N (+ N 1) 1)))))

prune793.0ms

Pruning

5 alts after pruning (1 fresh and 4 done)

Merged error: 0.2b

regimes67.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample1.7s

Algorithm
intervals