Average Error: 14.6 → 0.4
Time: 8.2s
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. Initial simplification14.6

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

  4. Applied diff-atan13.6

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

  5. Simplified0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2018362 +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: 4.9s)Debug log

start61.0ms

Algorithm
intervals

setup24.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 10.9b

localize21.0ms

Local error

Found 2 expressions with local error:

0.9b
(- (atan (+ 1 N)) (atan N))
0.0b
(atan (+ 1 N))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
2 → 38
Calls

2 calls. Slowest were:

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

series15.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

12.0ms
(- (atan (+ 1 N)) (atan N))
3.0ms
(atan (+ 1 N))

simplify420.0ms

Counts
32 → 44
Calls

32 calls. Slowest were:

42.0ms
(fma (* (cbrt (atan (+ 1 N))) (cbrt (atan (+ 1 N)))) (cbrt (atan (+ 1 N))) (- (* (cbrt (atan N)) (* (cbrt (atan N)) (cbrt (atan N))))))
36.0ms
(fma (sqrt (atan (+ 1 N))) (sqrt (atan (+ 1 N))) (- (* (sqrt (atan N)) (sqrt (atan N)))))
33.0ms
(fma 1 (atan (+ 1 N)) (- (* (atan N) 1)))

prune401.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize19.0ms

Local error

Found 1 expressions with local error:

0.1b
(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))

series4.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

4.0ms
(atan2 1 (fma N (+ N 1) 1))

simplify70.0ms

Counts
3 → 12
Calls

3 calls. Slowest were:

26.0ms
(atan2 1 (fma N (+ N 1) 1))
22.0ms
(atan2 1 (fma N (+ N 1) 1))
21.0ms
(atan2 1 (fma N (+ N 1) 1))

prune135.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.1b

localize11.0ms

Local error

Found 4 expressions with local error:

13.5b
(sqrt (atan2 1 (fma N (+ N 1) 1)))
13.5b
(sqrt (atan2 1 (fma N (+ N 1) 1)))
0.7b
(* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1))))
0.1b
(atan2 1 (fma N (+ N 1) 1))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

9.0ms
(* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1))))
2.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))
1.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))

series32.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

16.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))
12.0ms
(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))

simplify978.0ms

Counts
42 → 81
Calls

42 calls. Slowest were:

60.0ms
(* (* (* (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
(* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1))))
41.0ms
(sqrt (cbrt (atan2 1 (fma N (+ N 1) 1))))

prune904.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.1b

localize12.0ms

Local error

Found 4 expressions with local error:

13.9b
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
13.9b
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
13.9b
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
13.5b
(sqrt (atan2 1 (fma N (+ N 1) 1)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 47
Calls

4 calls. Slowest were:

3.0ms
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
3.0ms
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
3.0ms
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))

series550.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

199.0ms
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
194.0ms
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
146.0ms
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
11.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))

simplify443.0ms

Counts
19 → 59
Calls

19 calls. Slowest were:

39.0ms
(sqrt (* (cbrt (atan2 1 (fma N (+ N 1) 1))) (cbrt (atan2 1 (fma N (+ N 1) 1)))))
32.0ms
(pow (atan2 1 (fma N (+ N 1) 1)) 1/3)
32.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))

prune736.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.1b

regimes68.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms