Average Error: 31.4 → 31.4
Time: 30.3s
Precision: 64
Internal Precision: 128
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\frac{x - \sin x}{x - \tan x}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

    \[\frac{x - \sin x}{x - \tan x}\]

  2. Final simplification31.4

    \[\leadsto \frac{x - \sin x}{x - \tan x}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

Details

Time bar (total: 29.7s)Debug log

sample599.0ms

Algorithm
intervals

simplify13.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

13.0ms
(/ (- x (sin x)) (- x (tan x)))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 34.8b

localize41.0ms

Local error

Found 2 expressions with local error:

1.5b
(- x (tan x))
1.2b
(- x (sin x))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 22
Calls

2 calls. Slowest were:

2.0ms
(- x (tan x))
0.0ms
(- x (sin x))

series114.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

75.0ms
(- x (tan x))
39.0ms
(- x (sin x))

simplify247.0ms

Counts
6 → 28
Calls

6 calls. Slowest were:

149.0ms
(- (+ (* 1/3 (pow x 3)) (+ (* 2/15 (pow x 5)) (* 17/315 (pow x 7)))))
79.0ms
(- (+ (* 1/6 (pow x 3)) (* 1/5040 (pow x 7))) (* 1/120 (pow x 5)))
6.0ms
(- x (/ (sin x) (cos x)))

prune223.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 34.5b

localize17.0ms

Local error

Found 4 expressions with local error:

13.5b
(/ (- (* x x) (* (sin x) (sin x))) (+ x (sin x)))
1.5b
(- x (tan x))
1.2b
(- (* x x) (* (sin x) (sin x)))
0.2b
(* (sin x) (sin x))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

13.0ms
(/ (- (* x x) (* (sin x) (sin x))) (+ x (sin x)))
7.0ms
(* (sin x) (sin x))
4.0ms
(- (* x x) (* (sin x) (sin x)))

series209.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

88.0ms
(/ (- (* x x) (* (sin x) (sin x))) (+ x (sin x)))
57.0ms
(- x (tan x))
49.0ms
(- (* x x) (* (sin x) (sin x)))
15.0ms
(* (sin x) (sin x))

simplify2.6s

Counts
66 → 95
Calls

66 calls. Slowest were:

281.0ms
(/ (- (* x x) (* (sin x) (sin x))) (+ x (sin x)))
155.0ms
(- (+ (* 1/3 (pow x 3)) (+ (* 2/15 (pow x 5)) (* 17/315 (pow x 7)))))
128.0ms
(/ (- (* x x) (* (sin x) (sin x))) (sqrt (+ x (sin x))))

prune813.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 34.4b

localize34.0ms

Local error

Found 4 expressions with local error:

29.1b
(/ (- (* x x) (exp (log (* (sin x) (sin x))))) (+ x (sin x)))
17.7b
(log (* (sin x) (sin x)))
13.6b
(- (* x x) (exp (log (* (sin x) (sin x)))))
1.7b
(exp (log (* (sin x) (sin x))))

rewrite73.0ms

Algorithm
rewrite-expression-head
Counts
4 → 88
Calls

4 calls. Slowest were:

40.0ms
(/ (- (* x x) (exp (log (* (sin x) (sin x))))) (+ x (sin x)))
13.0ms
(exp (log (* (sin x) (sin x))))
12.0ms
(- (* x x) (exp (log (* (sin x) (sin x)))))

series239.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

86.0ms
(log (* (sin x) (sin x)))
78.0ms
(/ (- (* x x) (exp (log (* (sin x) (sin x))))) (+ x (sin x)))
50.0ms
(- (* x x) (exp (log (* (sin x) (sin x)))))
25.0ms
(exp (log (* (sin x) (sin x))))

simplify2.1s

Counts
78 → 100
Calls

78 calls. Slowest were:

175.0ms
(- (* (* x x) (* x x)) (* (exp (log (* (sin x) (sin x)))) (exp (log (* (sin x) (sin x))))))
110.0ms
(/ (* (* (- (* x x) (exp (log (* (sin x) (sin x))))) (- (* x x) (exp (log (* (sin x) (sin x)))))) (- (* x x) (exp (log (* (sin x) (sin x)))))) (* (* (+ x (sin x)) (+ x (sin x))) (+ x (sin x))))
94.0ms
(/ (* (cbrt (- (* x x) (exp (log (* (sin x) (sin x)))))) (cbrt (- (* x x) (exp (log (* (sin x) (sin x))))))) (* (cbrt (+ x (sin x))) (cbrt (+ x (sin x)))))

prune1.1s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 34.3b

localize33.0ms

Local error

Found 4 expressions with local error:

30.2b
(/ (- (* x x) (log (exp (* (sin x) (sin x))))) (+ x (sin x)))
2.1b
(log (exp (* (sin x) (sin x))))
1.5b
(- x (tan x))
0.9b
(- (* x x) (log (exp (* (sin x) (sin x)))))

rewrite35.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

20.0ms
(/ (- (* x x) (log (exp (* (sin x) (sin x))))) (+ x (sin x)))
8.0ms
(- (* x x) (log (exp (* (sin x) (sin x)))))
4.0ms
(log (exp (* (sin x) (sin x))))

series239.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

118.0ms
(/ (- (* x x) (log (exp (* (sin x) (sin x))))) (+ x (sin x)))
59.0ms
(- x (tan x))
47.0ms
(- (* x x) (log (exp (* (sin x) (sin x)))))
16.0ms
(log (exp (* (sin x) (sin x))))

simplify2.6s

Counts
63 → 91
Calls

63 calls. Slowest were:

279.0ms
(- (pow (* x x) 3) (pow (log (exp (* (sin x) (sin x)))) 3))
215.0ms
(- (* (* x x) (* x x)) (* (log (exp (* (sin x) (sin x)))) (log (exp (* (sin x) (sin x))))))
160.0ms
(/ (- (* x x) (log (exp (* (sin x) (sin x))))) (+ x (sin x)))

prune985.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 34.0b

regimes86.0ms

Accuracy

0% (0.8b remaining)

Error of 31.4b against oracle of 30.6b and baseline of 31.4b

bsearch1.0ms

end0.0ms

sample17.1s

Algorithm
intervals