Average Error: 13.5 → 0.3
Time: 49.5s
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(x + \frac{\tan y + \tan z}{1 - \tan z \cdot \tan y}\right) - \tan a\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.5

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  2. Using strategy rm

  3. Applied tan-sum0.2

    \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right)\]
  4. Using strategy rm

  5. Applied associate-+r-0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019007 
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 0) (<= 0.5884142 x 505.5909)) (or (<= -1.796658e+308 y -9.425585e-310) (<= 1.284938e-309 y 1.751224e+308)) (or (<= -1.776707e+308 z -8.599796e-310) (<= 3.293145e-311 z 1.725154e+308)) (or (<= -1.796658e+308 a -9.425585e-310) (<= 1.284938e-309 a 1.751224e+308)))
  (+ x (- (tan (+ y z)) (tan a))))

Details

Time bar (total: 48.8s)Debug log

sample519.0ms

Algorithm
intervals

simplify42.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

42.0ms
(+ x (- (tan (+ y z)) (tan a)))

prune37.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.7b

localize21.0ms

Local error

Found 3 expressions with local error:

3.6b
(tan (+ y z))
0.1b
(- (tan (+ y z)) (tan a))
0.1b
(+ x (- (tan (+ y z)) (tan a)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

5.0ms
(- (tan (+ y z)) (tan a))
3.0ms
(+ x (- (tan (+ y z)) (tan a)))
2.0ms
(tan (+ y z))

series328.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

139.0ms
(- (tan (+ y z)) (tan a))
131.0ms
(+ x (- (tan (+ y z)) (tan a)))
58.0ms
(tan (+ y z))

simplify542.0ms

Counts
21 → 47
Calls

21 calls. Slowest were:

253.0ms
(- (* (+ (tan y) (tan z)) (cos a)) (* (- 1 (* (tan y) (tan z))) (sin a)))
125.0ms
(* (- 1 (* (tan y) (tan z))) (cos a))
37.0ms
(- (+ (/ (sin (+ z y)) (cos (+ z y))) x) (/ (sin a) (cos a)))

prune556.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.1b

localize33.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (tan y) (tan z))
0.1b
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
0.1b
(/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z))))
0.1b
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))

rewrite84.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

33.0ms
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
29.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
13.0ms
(/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z))))

series1.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

493.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
451.0ms
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
265.0ms
(/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z))))
109.0ms
(* (tan y) (tan z))

simplify3.1s

Counts
71 → 97
Calls

71 calls. Slowest were:

335.0ms
(/ (* (* (+ (tan y) (tan z)) (+ (tan y) (tan z))) (+ (tan y) (tan z))) (* (* (- 1 (* (tan y) (tan z))) (- 1 (* (tan y) (tan z)))) (- 1 (* (tan y) (tan z)))))
296.0ms
(- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z))))))) (/ (sin a) (cos a)))
261.0ms
(- (+ (/ (sin z) (* (cos z) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) (+ (/ (sin y) (* (cos y) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))))) x)) (/ (sin a) (cos a)))

prune1.7s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize29.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (sin y) (tan z))
0.2b
(/ (* (sin y) (tan z)) (cos y))
0.1b
(- (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (tan a))
0.1b
(/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y))))

rewrite76.0ms

Algorithm
rewrite-expression-head
Counts
4 → 92
Calls

4 calls. Slowest were:

46.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (tan a))
19.0ms
(/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y))))
6.0ms
(/ (* (sin y) (tan z)) (cos y))

series938.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

441.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (tan a))
267.0ms
(/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y))))
137.0ms
(/ (* (sin y) (tan z)) (cos y))
93.0ms
(* (sin y) (tan z))

simplify4.9s

Counts
80 → 104
Calls

80 calls. Slowest were:

530.0ms
(/ (* (* (* (sin y) (sin y)) (sin y)) (* (* (tan z) (tan z)) (tan z))) (* (* (cos y) (cos y)) (cos y)))
461.0ms
(/ (* (* (* (sin y) (tan z)) (* (sin y) (tan z))) (* (sin y) (tan z))) (* (* (cos y) (cos y)) (cos y)))
367.0ms
(* (- 1 (/ (* (sin y) (tan z)) (cos y))) (cos a))

prune1.9s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 4 expressions with local error:

1.6b
(/ (- (* (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y))))) (* (tan a) (tan a))) (+ (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (tan a)))
0.4b
(* (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))))
0.3b
(* (tan a) (tan a))
0.3b
(* (sin y) (tan z))

rewrite167.0ms

Algorithm
rewrite-expression-head
Counts
4 → 151
Calls

4 calls. Slowest were:

113.0ms
(/ (- (* (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y))))) (* (tan a) (tan a))) (+ (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (tan a)))
35.0ms
(* (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))))
8.0ms
(* (tan a) (tan a))

series1.9s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.5s
(/ (- (* (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y))))) (* (tan a) (tan a))) (+ (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (tan a)))
275.0ms
(* (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))))
88.0ms
(* (sin y) (tan z))
41.0ms
(* (tan a) (tan a))

simplify14.2s

Counts
143 → 163
Calls

143 calls. Slowest were:

459.0ms
(* (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))))
400.0ms
(/ (- (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (tan a)) (+ (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (tan a)))
337.0ms
(* (* (* (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y))))) (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y))))) (* (* (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y)))) (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y))))) (/ (+ (tan y) (tan z)) (- 1 (/ (* (sin y) (tan z)) (cos y))))))

prune2.8s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

regimes383.0ms

Accuracy

0% (0.2b remaining)

Error of 0.3b against oracle of 0.1b and baseline of 0.3b

bsearch3.0ms

end0.0ms

sample13.2s

Algorithm
intervals