Average Error: 13.2 → 0.2
Time: 1.3m
Precision: 64
Internal Precision: 128
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \frac{\left(\frac{\tan y + \tan z}{1 - {\left(\tan z \cdot \tan y\right)}^{3}} \cdot \left(\left(\left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - \tan z \cdot \tan y\right) + 1\right) \cdot \left(\left(\tan z \cdot \sin y\right) \cdot \left(\tan z \cdot \sin y\right)\right) + \left(1 + {\left(\tan z \cdot \tan y\right)}^{3}\right) \cdot \left(\cos y \cdot \cos y\right)\right)\right) \cdot \cos a - \sin a \cdot \left(\left(\cos y \cdot \cos y\right) \cdot \left(\left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - \tan z \cdot \tan y\right) + 1\right)\right)}{\left(\left(\cos y \cdot \cos y\right) \cdot \cos a\right) \cdot \left(\left(1 - \tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)}\]

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 flip3--0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\frac{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}}{1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)}}} - \tan a\right)\]

  6. Applied associate-/r/0.2

    \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)\right)} - \tan a\right)\]

  7. Simplified0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \color{blue}{\left(\left(1 + \tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)} - \tan a\right)\]
  8. Using strategy rm

  9. Applied tan-quot0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(\left(1 + \tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right) - \color{blue}{\frac{\sin a}{\cos a}}\right)\]

  10. Applied tan-quot0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(\left(1 + \tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \color{blue}{\frac{\sin y}{\cos y}}\right)\right) - \frac{\sin a}{\cos a}\right)\]

  11. Applied associate-*r/0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(\left(1 + \tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right) \cdot \color{blue}{\frac{\tan z \cdot \sin y}{\cos y}}\right) - \frac{\sin a}{\cos a}\right)\]

  12. Applied tan-quot0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(\left(1 + \tan z \cdot \tan y\right) + \left(\tan z \cdot \color{blue}{\frac{\sin y}{\cos y}}\right) \cdot \frac{\tan z \cdot \sin y}{\cos y}\right) - \frac{\sin a}{\cos a}\right)\]

  13. Applied associate-*r/0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(\left(1 + \tan z \cdot \tan y\right) + \color{blue}{\frac{\tan z \cdot \sin y}{\cos y}} \cdot \frac{\tan z \cdot \sin y}{\cos y}\right) - \frac{\sin a}{\cos a}\right)\]

  14. Applied frac-times0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(\left(1 + \tan z \cdot \tan y\right) + \color{blue}{\frac{\left(\tan z \cdot \sin y\right) \cdot \left(\tan z \cdot \sin y\right)}{\cos y \cdot \cos y}}\right) - \frac{\sin a}{\cos a}\right)\]

  15. Applied flip3-+0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(\color{blue}{\frac{{1}^{3} + {\left(\tan z \cdot \tan y\right)}^{3}}{1 \cdot 1 + \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - 1 \cdot \left(\tan z \cdot \tan y\right)\right)}} + \frac{\left(\tan z \cdot \sin y\right) \cdot \left(\tan z \cdot \sin y\right)}{\cos y \cdot \cos y}\right) - \frac{\sin a}{\cos a}\right)\]

  16. Applied frac-add0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \color{blue}{\frac{\left({1}^{3} + {\left(\tan z \cdot \tan y\right)}^{3}\right) \cdot \left(\cos y \cdot \cos y\right) + \left(1 \cdot 1 + \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - 1 \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\left(\tan z \cdot \sin y\right) \cdot \left(\tan z \cdot \sin y\right)\right)}{\left(1 \cdot 1 + \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - 1 \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\cos y \cdot \cos y\right)}} - \frac{\sin a}{\cos a}\right)\]

  17. Applied associate-*r/0.2

    \[\leadsto x + \left(\color{blue}{\frac{\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(\left({1}^{3} + {\left(\tan z \cdot \tan y\right)}^{3}\right) \cdot \left(\cos y \cdot \cos y\right) + \left(1 \cdot 1 + \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - 1 \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\left(\tan z \cdot \sin y\right) \cdot \left(\tan z \cdot \sin y\right)\right)\right)}{\left(1 \cdot 1 + \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - 1 \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\cos y \cdot \cos y\right)}} - \frac{\sin a}{\cos a}\right)\]

  18. Applied frac-sub0.2

    \[\leadsto x + \color{blue}{\frac{\left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(\left({1}^{3} + {\left(\tan z \cdot \tan y\right)}^{3}\right) \cdot \left(\cos y \cdot \cos y\right) + \left(1 \cdot 1 + \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - 1 \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\left(\tan z \cdot \sin y\right) \cdot \left(\tan z \cdot \sin y\right)\right)\right)\right) \cdot \cos a - \left(\left(1 \cdot 1 + \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - 1 \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\cos y \cdot \cos y\right)\right) \cdot \sin a}{\left(\left(1 \cdot 1 + \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - 1 \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\cos y \cdot \cos y\right)\right) \cdot \cos a}}\]

  19. Simplified0.2

    \[\leadsto x + \frac{\left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(\left({1}^{3} + {\left(\tan z \cdot \tan y\right)}^{3}\right) \cdot \left(\cos y \cdot \cos y\right) + \left(1 \cdot 1 + \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - 1 \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\left(\tan z \cdot \sin y\right) \cdot \left(\tan z \cdot \sin y\right)\right)\right)\right) \cdot \cos a - \left(\left(1 \cdot 1 + \left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - 1 \cdot \left(\tan z \cdot \tan y\right)\right)\right) \cdot \left(\cos y \cdot \cos y\right)\right) \cdot \sin a}{\color{blue}{\left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) + \left(1 - \tan z \cdot \tan y\right)\right) \cdot \left(\left(\cos y \cdot \cos y\right) \cdot \cos a\right)}}\]

  20. Final simplification0.2

    \[\leadsto x + \frac{\left(\frac{\tan y + \tan z}{1 - {\left(\tan z \cdot \tan y\right)}^{3}} \cdot \left(\left(\left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - \tan z \cdot \tan y\right) + 1\right) \cdot \left(\left(\tan z \cdot \sin y\right) \cdot \left(\tan z \cdot \sin y\right)\right) + \left(1 + {\left(\tan z \cdot \tan y\right)}^{3}\right) \cdot \left(\cos y \cdot \cos y\right)\right)\right) \cdot \cos a - \sin a \cdot \left(\left(\cos y \cdot \cos y\right) \cdot \left(\left(\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right) - \tan z \cdot \tan y\right) + 1\right)\right)}{\left(\left(\cos y \cdot \cos y\right) \cdot \cos a\right) \cdot \left(\left(1 - \tan z \cdot \tan y\right) + \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right)}\]

Reproduce

herbie shell --seed 2018365 
(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: 1.0m)Debug log

start618.0ms

Algorithm
intervals

setup52.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 13.9b

localize19.0ms

Local error

Found 3 expressions with local error:

5.0b
(tan (+ y z))
0.2b
(- (tan (+ y z)) (tan a))
0.2b
(+ x (- (tan (+ y z)) (tan a)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

21.0ms
(- (tan (+ y z)) (tan a))
5.0ms
(+ x (- (tan (+ y z)) (tan a)))
3.0ms
(tan (+ y z))

series351.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

144.0ms
(- (tan (+ y z)) (tan a))
133.0ms
(+ x (- (tan (+ y z)) (tan a)))
73.0ms
(tan (+ y z))

simplify510.0ms

Counts
21 → 47
Calls

21 calls. Slowest were:

191.0ms
(- (* (+ (tan y) (tan z)) (cos a)) (* (- 1 (* (tan y) (tan z))) (sin a)))
112.0ms
(* (- 1 (* (tan y) (tan z))) (cos a))
31.0ms
(- (+ (/ (sin (+ z y)) (cos (+ z y))) x) (/ (sin a) (cos a)))

prune680.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.1b

localize24.0ms

Local error

Found 4 expressions with local error:

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

rewrite80.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

42.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
19.0ms
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
11.0ms
(/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z))))

series1.2s

Counts
4 → 12
Calls

4 calls. Slowest were:

420.0ms
(+ x (- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a)))
397.0ms
(- (/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z)))) (tan a))
245.0ms
(/ (+ (tan y) (tan z)) (- 1 (* (tan y) (tan z))))
114.0ms
(* (tan y) (tan z))

simplify3.5s

Counts
71 → 97
Calls

71 calls. Slowest were:

372.0ms
(+ (* 1/3 (* (pow z 3) y)) (+ (* 1/3 (* z (pow y 3))) (* z y)))
323.0ms
(/ (* (* (+ (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)))))
314.0ms
(- (+ (/ (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)))

prune1.6s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 4 expressions with local error:

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

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

16.0ms
(- (* (+ (tan y) (tan z)) (/ 1 (- 1 (* (tan y) (tan z))))) (tan a))
12.0ms
(* (+ (tan y) (tan z)) (/ 1 (- 1 (* (tan y) (tan z)))))
10.0ms
(+ x (- (* (+ (tan y) (tan z)) (/ 1 (- 1 (* (tan y) (tan z))))) (tan a)))

series1.2s

Counts
4 → 12
Calls

4 calls. Slowest were:

420.0ms
(+ x (- (* (+ (tan y) (tan z)) (/ 1 (- 1 (* (tan y) (tan z))))) (tan a)))
415.0ms
(- (* (+ (tan y) (tan z)) (/ 1 (- 1 (* (tan y) (tan z))))) (tan a))
268.0ms
(* (+ (tan y) (tan z)) (/ 1 (- 1 (* (tan y) (tan z)))))
101.0ms
(* (tan y) (tan z))

simplify8.6s

Counts
67 → 95
Calls

67 calls. Slowest were:

599.0ms
(* (* (* (+ (tan y) (tan z)) (+ (tan y) (tan z))) (+ (tan y) (tan z))) (* (* (/ 1 (- 1 (* (tan y) (tan z)))) (/ 1 (- 1 (* (tan y) (tan z))))) (/ 1 (- 1 (* (tan y) (tan z))))))
426.0ms
(- (* (* (- (* (tan y) (tan y)) (* (tan z) (tan z))) (/ 1 (- 1 (* (tan y) (tan z))))) (cos a)) (* (- (tan y) (tan z)) (sin a)))
390.0ms
(+ (* 1/3 (* (pow z 3) y)) (+ (* 1/3 (* z (pow y 3))) (* z y)))

prune1.7s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize63.0ms

Local error

Found 4 expressions with local error:

0.3b
(pow (* (tan y) (tan z)) 3)
0.3b
(* (* (tan z) (tan y)) (* (tan z) (tan y)))
0.3b
(- (* (/ (+ (tan y) (tan z)) (- (pow 1 3) (pow (* (tan y) (tan z)) 3))) (+ (+ 1 (* (tan z) (tan y))) (* (* (tan z) (tan y)) (* (tan z) (tan y))))) (tan a))
0.2b
(* (tan y) (tan z))

rewrite422.0ms

Algorithm
rewrite-expression-head
Counts
4 → 164
Calls

4 calls. Slowest were:

388.0ms
(- (* (/ (+ (tan y) (tan z)) (- (pow 1 3) (pow (* (tan y) (tan z)) 3))) (+ (+ 1 (* (tan z) (tan y))) (* (* (tan z) (tan y)) (* (tan z) (tan y))))) (tan a))
15.0ms
(* (* (tan z) (tan y)) (* (tan z) (tan y)))
10.0ms
(pow (* (tan y) (tan z)) 3)

series2.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.9s
(- (* (/ (+ (tan y) (tan z)) (- (pow 1 3) (pow (* (tan y) (tan z)) 3))) (+ (+ 1 (* (tan z) (tan y))) (* (* (tan z) (tan y)) (* (tan z) (tan y))))) (tan a))
146.0ms
(pow (* (tan y) (tan z)) 3)
117.0ms
(* (* (tan z) (tan y)) (* (tan z) (tan y)))
91.0ms
(* (tan y) (tan z))

simplify33.6s

Counts
216 → 176
Calls

216 calls. Slowest were:

747.0ms
(* (* (* (* (tan z) (tan z)) (tan z)) (* (* (tan y) (tan y)) (tan y))) (* (* (* (tan z) (tan y)) (* (tan z) (tan y))) (* (tan z) (tan y))))
648.0ms
(* (* (* (* (tan z) (tan y)) (* (tan z) (tan y))) (* (tan z) (tan y))) (* (* (* (tan z) (tan y)) (* (tan z) (tan y))) (* (tan z) (tan y))))
597.0ms
(* (* (* (* (tan z) (tan y)) (* (tan z) (tan y))) (* (tan z) (tan y))) (* (* (* (tan z) (tan z)) (tan z)) (* (* (tan y) (tan y)) (tan y))))

prune3.3s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

regimes559.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.1b and baseline of 0.2b

bsearch8.0ms