Average Error: 15.3 → 0.4
Time: 7.8s
Precision: 64
Internal Precision: 128
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{1 + \left(N + {N}^{2}\right)}\]

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.4
Herbie0.4
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 15.3

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

  3. Applied diff-atan14.2

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

  4. Simplified0.4

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

  5. Taylor expanded around inf 0.4

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \color{blue}{\left(N + {N}^{2}\right)}}\]

  6. Final simplification0.4

    \[\leadsto \tan^{-1}_* \frac{1}{1 + \left(N + {N}^{2}\right)}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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

Details

Time bar (total: 7.2s)Debug log

sample71.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 14.9b

localize22.0ms

Local error

Found 2 expressions with local error:

1.2b
(- (atan (+ N 1)) (atan N))
0.0b
(atan (+ N 1))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 22
Calls

2 calls. Slowest were:

3.0ms
(- (atan (+ N 1)) (atan N))
1.0ms
(atan (+ N 1))

series8.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

7.0ms
(- (atan (+ N 1)) (atan N))
2.0ms
(atan (+ N 1))

simplify537.0ms

Counts
12 → 28
Calls

12 calls. Slowest were:

459.0ms
(- (+ N 1) N)
36.0ms
(+ 1 (* (+ N 1) N))
13.0ms
(/ (exp (atan (+ N 1))) (exp (atan N)))

prune192.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize8.0ms

Local error

Found 2 expressions with local error:

0.2b
(atan2 1 (+ 1 (* (+ N 1) N)))
0.0b
(* (+ N 1) N)

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
2 → 20
Calls

2 calls. Slowest were:

3.0ms
(* (+ N 1) N)
0.0ms
(atan2 1 (+ 1 (* (+ N 1) N)))

series30.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

27.0ms
(* (+ N 1) N)
2.0ms
(atan2 1 (+ 1 (* (+ N 1) N)))

simplify196.0ms

Counts
10 → 26
Calls

10 calls. Slowest were:

99.0ms
(* (- (* N N) (* 1 1)) N)
27.0ms
(* (+ (pow N 3) (pow 1 3)) N)
26.0ms
(atan2 1 (+ N (+ (pow N 2) 1)))

prune233.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.1b

localize12.0ms

Local error

Found 2 expressions with local error:

0.2b
(atan2 1 (+ 1 (+ N (pow N 2))))
0.0b
(+ N (pow N 2))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
2 → 18
Calls

2 calls. Slowest were:

1.0ms
(+ N (pow N 2))
0.0ms
(atan2 1 (+ 1 (+ N (pow N 2))))

series20.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

18.0ms
(+ N (pow N 2))
3.0ms
(atan2 1 (+ 1 (+ N (pow N 2))))

simplify65.0ms

Counts
6 → 24
Calls

6 calls. Slowest were:

24.0ms
(atan2 1 (+ N (+ (pow N 2) 1)))
19.0ms
(atan2 1 (+ N (+ (pow N 2) 1)))
12.0ms
(atan2 1 (+ N (+ (pow N 2) 1)))

prune234.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.1b

localize22.0ms

Local error

Found 4 expressions with local error:

15.0b
(cbrt (* (+ N 1) N))
15.0b
(cbrt (* (+ N 1) N))
15.0b
(cbrt (* (+ N 1) N))
0.5b
(* (cbrt (* (+ N 1) N)) (cbrt (* (+ N 1) N)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

9.0ms
(* (cbrt (* (+ N 1) N)) (cbrt (* (+ N 1) N)))
2.0ms
(cbrt (* (+ N 1) N))
1.0ms
(cbrt (* (+ N 1) N))

series211.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

61.0ms
(* (cbrt (* (+ N 1) N)) (cbrt (* (+ N 1) N)))
59.0ms
(cbrt (* (+ N 1) N))
51.0ms
(cbrt (* (+ N 1) N))
40.0ms
(cbrt (* (+ N 1) N))

simplify2.9s

Counts
49 → 77
Calls

49 calls. Slowest were:

387.0ms
(* (* (+ N 1) N) (* (+ N 1) N))
311.0ms
(* (* (+ N 1) N) (* (+ N 1) N))
234.0ms
(- (+ (* (pow (pow N 4) 1/3) (pow (cbrt -1) 2)) (* 2/3 (* (pow N 1/3) (pow (cbrt -1) 2)))) (* 1/9 (* (pow (/ 1 (pow N 2)) 1/3) (pow (cbrt -1) 2))))

prune670.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.1b

regimes40.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample1.7s

Algorithm
intervals