Average Error: 13.2 → 0.3
Time: 34.6s
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\log \left(e^{\left(\frac{\tan y + \tan z}{1 - \tan z \cdot \tan y} - \tan a\right) + x}\right)\]

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 13.2

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 add-log-exp0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019004 
(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: 33.3s)Debug log

sample577.0ms

Algorithm
intervals

simplify31.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

31.0ms
(+ x (- (tan (+ y z)) (tan a)))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 16.4b

localize36.0ms

Local error

Found 3 expressions with local error:

3.7b
(tan (+ y z))
0.2b
(- (tan (+ y z)) (tan a))
0.1b
(+ x (- (tan (+ y z)) (tan a)))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

10.0ms
(- (tan (+ y z)) (tan a))
5.0ms
(+ x (- (tan (+ y z)) (tan a)))
3.0ms
(tan (+ y z))

series269.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

114.0ms
(+ x (- (tan (+ y z)) (tan a)))
103.0ms
(- (tan (+ y z)) (tan a))
51.0ms
(tan (+ y z))

simplify467.0ms

Counts
21 → 47
Calls

21 calls. Slowest were:

177.0ms
(- (* (+ (tan y) (tan z)) (cos a)) (* (- 1 (* (tan y) (tan z))) (sin a)))
132.0ms
(* (- 1 (* (tan y) (tan z))) (cos a))
32.0ms
(- (+ (/ (sin (+ z y)) (cos (+ z y))) x) (/ (sin a) (cos a)))

prune561.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.1b

localize38.0ms

Local error

Found 4 expressions with local error:

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

rewrite81.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

28.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
24.0ms
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
22.0ms
(/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z))))

series1.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

367.0ms
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
360.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
274.0ms
(/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z))))
101.0ms
(* (tan y) (tan z))

simplify3.0s

Counts
71 → 97
Calls

71 calls. Slowest were:

346.0ms
(+ (* 1/3 (* (pow z 3) y)) (+ (* 1/3 (* z (pow y 3))) (* z y)))
289.0ms
(/ (* (* (+ (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)))))
264.0ms
(- (+ (/ (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)))

prune1.4s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.1b

localize63.0ms

Local error

Found 4 expressions with local error:

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

rewrite129.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

105.0ms
(- (/ (+ (/ (sin z) (cos z)) (/ (sin y) (cos y))) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z))))) (tan a))
13.0ms
(/ (* (sin z) (sin y)) (* (cos y) (cos z)))
6.0ms
(* (sin z) (sin y))

series530.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

330.0ms
(- (/ (+ (/ (sin z) (cos z)) (/ (sin y) (cos y))) (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z))))) (tan a))
116.0ms
(/ (* (sin z) (sin y)) (* (cos y) (cos z)))
64.0ms
(* (sin z) (sin y))
19.0ms
(/ (sin y) (cos y))

simplify3.6s

Counts
67 → 97
Calls

67 calls. Slowest were:

365.0ms
(* (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))) (cos a))
296.0ms
(/ (* (* (* (sin z) (sin z)) (sin z)) (* (* (sin y) (sin y)) (sin y))) (* (* (* (cos y) (cos y)) (cos y)) (* (* (cos z) (cos z)) (cos z))))
273.0ms
(- (+ (/ (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)))

prune2.0s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.1b

localize35.0ms

Local error

Found 4 expressions with local error:

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

rewrite119.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

87.0ms
(- (/ (+ (* (sin y) (cos z)) (* (cos y) (sin z))) (* (- 1 (* (tan y) (tan z))) (* (cos y) (cos z)))) (tan a))
23.0ms
(/ (+ (* (sin y) (cos z)) (* (cos y) (sin z))) (* (- 1 (* (tan y) (tan z))) (* (cos y) (cos z))))
4.0ms
(* (tan y) (tan z))

series764.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

376.0ms
(- (/ (+ (* (sin y) (cos z)) (* (cos y) (sin z))) (* (- 1 (* (tan y) (tan z))) (* (cos y) (cos z)))) (tan a))
254.0ms
(/ (+ (* (sin y) (cos z)) (* (cos y) (sin z))) (* (- 1 (* (tan y) (tan z))) (* (cos y) (cos z))))
89.0ms
(* (tan y) (tan z))
44.0ms
(* (cos y) (sin z))

simplify4.8s

Counts
59 → 94
Calls

59 calls. Slowest were:

311.0ms
(+ (* 1/3 (* (pow z 3) y)) (+ (* 1/3 (* z (pow y 3))) (* z y)))
282.0ms
(* (+ (* 1 1) (+ (* (* (tan y) (tan z)) (* (tan y) (tan z))) (* 1 (* (tan y) (tan z))))) 2)
261.0ms
(/ (+ (* (cos z) (sin y)) (* (sin z) (cos y))) (* (cos z) (* (- 1 (/ (* (sin z) (sin y)) (* (cos y) (cos z)))) (cos y))))

prune1.7s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.1b

regimes469.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms

end0.0ms

sample11.6s

Algorithm
intervals