Average Error: 14.9 → 0.3
Time: 6.8s
Precision: 64
Internal Precision: 128
\[\tan^{-1} \left(N + 1\right) - \tan^{-1} N\]
↓
\[\tan^{-1}_* \frac{1}{\left(N + 1\right) \cdot N + 1}\]

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.3
Herbie0.3
\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right)\]

Derivation

  1. Initial program 14.9

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

  3. Applied diff-atan13.8

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019010 
(FPCore (N)
  :name "2atan (example 3.5)"

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

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

Details

Time bar (total: 6.1s)Debug log

sample64.0ms

Algorithm
intervals

simplify7.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
7.0ms
(- (atan (+ N 1)) (atan N))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 16.1b

localize21.0ms

Local error

Found 2 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×*-un-lft-identity
4×add-sqr-sqrt
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×pow1
1×difference-of-squares
1×distribute-lft-out--
1×flip--
1×diff-atan
1×diff-log
1×flip3--
1×sub-neg
Counts
2 → 22
Calls
2 calls:
Slowest
6.0ms
(- (atan (+ N 1)) (atan N))
1.0ms
(atan (+ N 1))

series12.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
10.0ms
(- (atan (+ N 1)) (atan N))
2.0ms
(atan (+ N 1))

simplify114.0ms

Counts
12 → 28
Calls
12 calls:
Slowest
30.0ms
(+ 1 (* (+ N 1) N))
23.0ms
(/ (exp (atan (+ N 1))) (exp (atan N)))
8.0ms
(- (sqrt (atan (+ N 1))) (sqrt (atan N)))
8.0ms
(- (+ N 1) N)
7.0ms
(- (atan (+ N 1)) (atan N))

prune198.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.2b

localize8.0ms

Local error

Found 2 expressions with local error:

0.3b
(atan2 1 (+ 1 (* (+ N 1) N)))
0.0b
(* (+ N 1) N)

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
3×add-cube-cbrt
3×associate-*l*
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
2×add-exp-log
2×associate-*l/
2×add-cbrt-cube
2×pow1
1×flip-+
1×*-commutative
1×flip3-+
Counts
2 → 20
Calls
2 calls:
Slowest
2.0ms
(* (+ N 1) N)
0.0ms
(atan2 1 (+ 1 (* (+ N 1) N)))

series31.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
28.0ms
(* (+ N 1) N)
3.0ms
(atan2 1 (+ 1 (* (+ N 1) N)))

simplify183.0ms

Counts
10 → 26
Calls
10 calls:
Slowest
92.0ms
(* (- (* N N) (* 1 1)) N)
25.0ms
(* (+ (pow N 3) (pow 1 3)) N)
20.0ms
(atan2 1 (+ N (+ (pow N 2) 1)))
14.0ms
(atan2 1 (+ N (+ (pow N 2) 1)))
13.0ms
(atan2 1 (+ N (+ (pow N 2) 1)))

prune181.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.2b

localize6.0ms

Local error

Found 4 expressions with local error:

12.9b
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
12.9b
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
0.6b
(* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))
0.3b
(atan2 1 (+ 1 (* (+ N 1) N)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
12×sqrt-prod
12×add-sqr-sqrt
10×add-cube-cbrt
10×*-un-lft-identity
9×pow1
6×add-exp-log
6×associate-*l*
6×associate-*r*
6×pow1/2
6×add-cbrt-cube
4×add-log-exp
2×pow-prod-up
2×pow-prod-down
2×rem-sqrt-square
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×sqrt-unprod
1×pow-plus
1×rem-square-sqrt
1×pow2
Counts
4 → 61
Calls
4 calls:
Slowest
8.0ms
(* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))
2.0ms
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
1.0ms
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
0.0ms
(atan2 1 (+ 1 (* (+ N 1) N)))

series40.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
21.0ms
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
14.0ms
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
3.0ms
(* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))
2.0ms
(atan2 1 (+ 1 (* (+ N 1) N)))

simplify1.1s

Counts
42 → 73
Calls
42 calls:
Slowest
46.0ms
(sqrt (cbrt (atan2 1 (+ 1 (* (+ N 1) N)))))
45.0ms
(* (* (* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (* (* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))))
43.0ms
(sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))
43.0ms
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
43.0ms
(cbrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))

prune748.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.2b

localize9.0ms

Local error

Found 4 expressions with local error:

12.9b
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
12.9b
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
12.9b
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
0.6b
(* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (* (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
12×sqrt-prod
12×add-sqr-sqrt
10×pow1
9×add-cube-cbrt
9×add-exp-log
9×add-cbrt-cube
9×*-un-lft-identity
8×pow1/2
6×associate-*l*
5×pow-prod-down
4×add-log-exp
3×cbrt-unprod
3×prod-exp
3×rem-sqrt-square
2×pow-prod-up
2×sqrt-unprod
1×*-commutative
1×associate-*r*
Counts
4 → 60
Calls
4 calls:
Slowest
21.0ms
(* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (* (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))))
1.0ms
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
0.0ms
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
0.0ms
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))

series51.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
22.0ms
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
13.0ms
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
12.0ms
(sqrt (atan2 1 (+ 1 (* (+ N 1) N))))
3.0ms
(* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (* (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))))

simplify1.2s

Counts
40 → 72
Calls
40 calls:
Slowest
107.0ms
(* (* (* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (* (* (* (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))) (* (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))))) (* (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))))))
103.0ms
(* (* (* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (* (* (* (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))) (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))) (* (* (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))) (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))))))
54.0ms
(* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (* (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))))
54.0ms
(+ (log (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))) (+ (log (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))))) (log (sqrt (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))))))
45.0ms
(* (atan2 1 (+ 1 (* (+ N 1) N))) (* (sqrt (atan2 1 (+ 1 (* (+ N 1) N)))) (sqrt (atan2 1 (+ 1 (* (+ N 1) N))))))

prune621.0ms

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0.2b

regimes31.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample1.5s

Algorithm
intervals