Average Error: 14.9 → 0.4
Time: 5.5s
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

Original14.9
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

    \[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]

  2. Initial simplification14.9

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

  4. Applied diff-atan13.8

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

  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 2018365 +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: 3.4s)Debug log

start71.0ms

Algorithm
intervals

setup17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 14.8b

localize10.0ms

Local error

Found 1 expressions with local error:

1.1b
(- (atan (+ 1 N)) (atan N))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

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

series11.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

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

simplify408.0ms

Counts
29 → 32
Calls

29 calls. Slowest were:

31.0ms
(fma 1 (atan (+ 1 N)) (- (* (atan N) 1)))
28.0ms
(fma (- (cbrt (atan N))) (* (cbrt (atan N)) (cbrt (atan N))) (* (cbrt (atan N)) (* (cbrt (atan N)) (cbrt (atan N)))))
25.0ms
(fma (- (cbrt (atan N))) (* (cbrt (atan N)) (cbrt (atan N))) (* (cbrt (atan N)) (* (cbrt (atan N)) (cbrt (atan N)))))

prune237.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.5b

localize18.0ms

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))

series4.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

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

simplify39.0ms

Counts
3 → 12
Calls

3 calls. Slowest were:

16.0ms
(atan2 1 (fma N (+ N 1) 1))
12.0ms
(atan2 1 (fma N (+ N 1) 1))
11.0ms
(atan2 1 (fma N (+ N 1) 1))

prune90.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.5b

localize11.0ms

Local error

Found 4 expressions with local error:

17.5b
(sqrt (atan2 1 (fma N (+ N 1) 1)))
17.5b
(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))))
0.5b
(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)))

series33.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

18.0ms
(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))

simplify677.0ms

Counts
42 → 81
Calls

42 calls. Slowest were:

37.0ms
(sqrt (sqrt (atan2 1 (fma N (+ N 1) 1))))
33.0ms
(* (* (* (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)))))
30.0ms
(sqrt (atan2 1 (fma N (+ N 1) 1)))

prune543.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.4b

localize13.0ms

Local error

Found 4 expressions with local error:

18.1b
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
18.1b
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
18.1b
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
17.5b
(sqrt (atan2 1 (fma N (+ N 1) 1)))

rewrite9.0ms

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)))))

series400.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

131.0ms
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
129.0ms
(cbrt (* (sqrt (atan2 1 (fma N (+ N 1) 1))) (sqrt (atan2 1 (fma N (+ N 1) 1)))))
129.0ms
(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)))

simplify307.0ms

Counts
19 → 59
Calls

19 calls. Slowest were:

20.0ms
(sqrt (* (cbrt (atan2 1 (fma N (+ N 1) 1))) (cbrt (atan2 1 (fma N (+ N 1) 1)))))
19.0ms
(cbrt (sqrt (atan2 1 (fma N (+ N 1) 1))))
19.0ms
(cbrt (sqrt (atan2 1 (fma N (+ N 1) 1))))

prune487.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.4b

regimes30.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms