Average Error: 0.3 → 0.4
Time: 19.0s
Precision: 64
Internal Precision: 128
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{\tan x \cdot \tan x + 1} \cdot \left(1 - \tan x \cdot \tan x\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.3

    \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
  2. Using strategy rm

  3. Applied div-sub0.4

    \[\leadsto \color{blue}{\frac{1}{1 + \tan x \cdot \tan x} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}}\]
  4. Using strategy rm

  5. Applied div-inv0.4

    \[\leadsto \frac{1}{1 + \tan x \cdot \tan x} - \color{blue}{\left(\tan x \cdot \tan x\right) \cdot \frac{1}{1 + \tan x \cdot \tan x}}\]

  6. Applied *-un-lft-identity0.4

    \[\leadsto \color{blue}{1 \cdot \frac{1}{1 + \tan x \cdot \tan x}} - \left(\tan x \cdot \tan x\right) \cdot \frac{1}{1 + \tan x \cdot \tan x}\]

  7. Applied distribute-rgt-out--0.4

    \[\leadsto \color{blue}{\frac{1}{1 + \tan x \cdot \tan x} \cdot \left(1 - \tan x \cdot \tan x\right)}\]

  8. Final simplification0.4

    \[\leadsto \frac{1}{\tan x \cdot \tan x + 1} \cdot \left(1 - \tan x \cdot \tan x\right)\]

Reproduce

herbie shell --seed 2019004 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))

Details

Time bar (total: 17.9s)Debug log

sample290.0ms

Algorithm
intervals

simplify83.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

83.0ms
(/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x))))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize33.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (tan x) (tan x))
0.3b
(* (tan x) (tan x))
0.2b
(/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x))))
0.2b
(- 1 (* (tan x) (tan x)))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 89
Calls

4 calls. Slowest were:

10.0ms
(* (tan x) (tan x))
8.0ms
(* (tan x) (tan x))
8.0ms
(/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x))))

series187.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

88.0ms
(/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x))))
45.0ms
(* (tan x) (tan x))
31.0ms
(* (tan x) (tan x))
23.0ms
(- 1 (* (tan x) (tan x)))

simplify2.0s

Counts
72 → 101
Calls

72 calls. Slowest were:

480.0ms
(/ (* (* (- 1 (* (tan x) (tan x))) (- 1 (* (tan x) (tan x)))) (- 1 (* (tan x) (tan x)))) (* (* (+ 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))) (+ 1 (* (tan x) (tan x)))))
263.0ms
(/ (- 1 (/ (pow (sin x) 2) (pow (cos x) 2))) (+ (/ (pow (sin x) 2) (pow (cos x) 2)) 1))
234.0ms
(+ (* 1 1) (- (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* 1 (* (tan x) (tan x)))))

prune942.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.0b

localize30.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
0.3b
(* (tan x) (tan x))
0.3b
(* (tan x) (tan x))
0.3b
(* (tan x) (tan x))

rewrite56.0ms

Algorithm
rewrite-expression-head
Counts
4 → 127
Calls

4 calls. Slowest were:

34.0ms
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
8.0ms
(* (tan x) (tan x))
8.0ms
(* (tan x) (tan x))

series129.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

39.0ms
(* (tan x) (tan x))
33.0ms
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
32.0ms
(* (tan x) (tan x))
25.0ms
(* (tan x) (tan x))

simplify1.0s

Counts
132 → 139
Calls

132 calls. Slowest were:

126.0ms
(* (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))) (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))))
104.0ms
(* (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))) (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))))
91.0ms
(* (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))) (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))))

prune1.4s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
0.3b
(* (tan x) (tan x))
0.3b
(* (tan x) (tan x))
0.3b
(* (tan x) (tan x))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 127
Calls

4 calls. Slowest were:

17.0ms
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
8.0ms
(* (tan x) (tan x))
8.0ms
(* (tan x) (tan x))

series161.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

44.0ms
(* (tan x) (tan x))
44.0ms
(* (tan x) (tan x))
41.0ms
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
32.0ms
(* (tan x) (tan x))

simplify975.0ms

Counts
132 → 139
Calls

132 calls. Slowest were:

147.0ms
(* (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))) (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))))
95.0ms
(* (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))) (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))))
85.0ms
(* (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))) (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))))

prune1.3s

Pruning

14 alts after pruning (13 fresh and 1 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

0.4b
(- (/ 1 (+ 1 (* (tan x) (tan x)))) (/ (* (tan x) (tan x)) (+ 1 (* (tan x) (tan x)))))
0.3b
(* (tan x) (tan x))
0.3b
(* (tan x) (tan x))
0.3b
(* (tan x) (tan x))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 91
Calls

4 calls. Slowest were:

12.0ms
(- (/ 1 (+ 1 (* (tan x) (tan x)))) (/ (* (tan x) (tan x)) (+ 1 (* (tan x) (tan x)))))
12.0ms
(* (tan x) (tan x))
8.0ms
(* (tan x) (tan x))

series230.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

131.0ms
(- (/ 1 (+ 1 (* (tan x) (tan x)))) (/ (* (tan x) (tan x)) (+ 1 (* (tan x) (tan x)))))
36.0ms
(* (tan x) (tan x))
34.0ms
(* (tan x) (tan x))
28.0ms
(* (tan x) (tan x))

simplify1.6s

Counts
76 → 103
Calls

76 calls. Slowest were:

399.0ms
(- (* 1 (+ 1 (* (tan x) (tan x)))) (* (+ 1 (* (tan x) (tan x))) (* (tan x) (tan x))))
246.0ms
(+ (* 1 1) (- (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* 1 (* (tan x) (tan x)))))
155.0ms
(- (/ 1 (+ (/ (pow (sin x) 2) (pow (cos x) 2)) 1)) (/ (pow (sin x) 2) (* (pow (cos x) 2) (+ (/ (pow (sin x) 2) (pow (cos x) 2)) 1))))

prune995.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

regimes223.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.1b and baseline of 0.4b

bsearch2.0ms

end0.0ms

sample6.1s

Algorithm
intervals