Average Error: 37.0 → 0.5
Time: 45.5s
Precision: 64
Internal Precision: 128
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right)} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right)}{1 - \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right)}\right) + \frac{\left(\sin x - \sin x\right) - \left(-\sin x\right) \cdot \left(\left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}\right)\right)}{\cos x \cdot \left(1 - \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right)\right)}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target14.7
Herbie0.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

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

  3. Applied tan-sum22.3

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

  5. Applied flip--22.3

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

  6. Taylor expanded around -inf 22.4

    \[\leadsto \color{blue}{\left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)} + \left(\frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\cos x \cdot \left({\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)\right)} + \left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)\right)} + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right)}\right)\right)\right) - \frac{\sin x}{\cos x}}\]

  7. Simplified0.6

    \[\leadsto \color{blue}{\left(\frac{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right)} + \frac{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right)}\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)}\]
  8. Using strategy rm

  9. Applied frac-sub2.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2019010 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))

Details

Time bar (total: 44.7s)Debug log

sample438.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
8.0ms
(- (tan (+ x eps)) (tan x))

prune49.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 35.1b

localize19.0ms

Local error

Found 2 expressions with local error:

2.9b
(tan (+ x eps))
1.8b
(- (tan (+ x eps)) (tan x))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×tan-quot
4×*-un-lft-identity
4×add-sqr-sqrt
2×add-cube-cbrt
2×frac-sub
2×add-exp-log
2×add-cbrt-cube
2×tan-sum
2×pow1
1×difference-of-squares
1×distribute-lft-out--
1×flip--
1×diff-log
1×flip3--
1×sub-neg
Counts
2 → 25
Calls
2 calls:
Slowest
5.0ms
(- (tan (+ x eps)) (tan x))
1.0ms
(tan (+ x eps))

series174.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
108.0ms
(- (tan (+ x eps)) (tan x))
66.0ms
(tan (+ x eps))

simplify640.0ms

Counts
15 → 31
Calls
15 calls:
Slowest
208.0ms
(- (* (+ (tan x) (tan eps)) (cos x)) (* (- 1 (* (tan x) (tan eps))) (sin x)))
169.0ms
(* (- 1 (* (tan x) (tan eps))) (cos x))
139.0ms
(+ (* x (pow eps 2)) (+ eps (* (pow x 2) eps)))
48.0ms
(+ x (+ (* 1/3 (pow x 3)) eps))
17.0ms
(/ (exp (tan (+ x eps))) (exp (tan x)))

prune314.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 16.6b

localize35.0ms

Local error

Found 4 expressions with local error:

4.1b
(- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x))
0.2b
(* (tan x) (tan eps))
0.1b
(+ (tan x) (tan eps))
0.1b
(- 1 (* (tan x) (tan eps)))

rewrite67.0ms

Algorithm
rewrite-expression-head
Rules
11×add-sqr-sqrt
10×*-un-lft-identity
8×add-log-exp
7×tan-quot
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×pow1
3×associate-*l*
3×associate-*r*
2×difference-of-squares
2×flip--
2×flip3--
2×sub-neg
1×distribute-lft-out--
1×distribute-lft-out
1×flip-+
1×cbrt-unprod
1×frac-sub
1×*-commutative
1×associate-*r/
1×prod-exp
1×frac-add
1×associate-*l/
1×pow-prod-down
1×diff-log
1×times-frac
1×flip3-+
1×sum-log
1×+-commutative
1×frac-times
Counts
4 → 60
Calls
4 calls:
Slowest
49.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x))
8.0ms
(* (tan x) (tan eps))
7.0ms
(+ (tan x) (tan eps))
2.0ms
(- 1 (* (tan x) (tan eps)))

series810.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
398.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x))
162.0ms
(- 1 (* (tan x) (tan eps)))
136.0ms
(+ (tan x) (tan eps))
113.0ms
(* (tan x) (tan eps))

simplify2.3s

Counts
36 → 72
Calls
36 calls:
Slowest
365.0ms
(- 1 (+ (* 1/3 (* (pow x 3) eps)) (* x eps)))
334.0ms
(+ (* 1/3 (* (pow x 3) eps)) (+ (* 1/3 (* x (pow eps 3))) (* x eps)))
267.0ms
(- (+ (/ (sin eps) (* (cos eps) (- 1 (/ (* (sin x) (sin eps)) (* (cos x) (cos eps)))))) (/ (sin x) (* (cos x) (- 1 (/ (* (sin x) (sin eps)) (* (cos x) (cos eps))))))) (/ (sin x) (cos x)))
241.0ms
(- (+ (/ (sin eps) (* (cos eps) (- 1 (/ (* (sin x) (sin eps)) (* (cos x) (cos eps)))))) (/ (sin x) (* (cos x) (- 1 (/ (* (sin x) (sin eps)) (* (cos x) (cos eps))))))) (/ (sin x) (cos x)))
168.0ms
(* (- 1 (* (tan x) (tan eps))) (cos x))

prune829.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 16.5b

localize40.0ms

Local error

Found 4 expressions with local error:

4.1b
(- (/ (+ (tan x) (tan eps)) (/ (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps)))) (+ 1 (* (tan x) (tan eps))))) (tan x))
0.3b
(* (* (tan x) (tan eps)) (* (tan x) (tan eps)))
0.2b
(* (tan x) (tan eps))
0.2b
(* (tan x) (tan eps))

rewrite115.0ms

Algorithm
rewrite-expression-head
Rules
41×tan-quot
35×pow1
20×add-exp-log
20×add-cbrt-cube
19×frac-times
17×add-sqr-sqrt
15×pow-prod-down
13×associate-*r/
13×associate-*l/
10×cbrt-unprod
10×prod-exp
10×*-un-lft-identity
8×add-cube-cbrt
7×associate-*l*
7×associate-*r*
6×add-log-exp
4×pow-prod-up
3×difference-of-squares
3×*-commutative
3×times-frac
2×pow-plus
1×distribute-lft-out--
1×flip--
1×frac-sub
1×diff-log
1×flip3--
1×sub-neg
1×pow2
Counts
4 → 101
Calls
4 calls:
Slowest
68.0ms
(- (/ (+ (tan x) (tan eps)) (/ (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps)))) (+ 1 (* (tan x) (tan eps))))) (tan x))
36.0ms
(* (* (tan x) (tan eps)) (* (tan x) (tan eps)))
6.0ms
(* (tan x) (tan eps))
4.0ms
(* (tan x) (tan eps))

series1.3s

Counts
4 → 12
Calls
4 calls:
Slowest
984.0ms
(- (/ (+ (tan x) (tan eps)) (/ (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps)))) (+ 1 (* (tan x) (tan eps))))) (tan x))
119.0ms
(* (* (tan x) (tan eps)) (* (tan x) (tan eps)))
109.0ms
(* (tan x) (tan eps))
108.0ms
(* (tan x) (tan eps))

simplify7.2s

Counts
94 → 113
Calls
94 calls:
Slowest
648.0ms
(* (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan eps) (tan eps)) (tan eps))) (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))))
614.0ms
(* (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))) (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))))
565.0ms
(* (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))) (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan eps) (tan eps)) (tan eps))))
417.0ms
(/ (* (pow (sin x) 2) (pow (sin eps) 2)) (* (pow (cos x) 2) (pow (cos eps) 2)))
413.0ms
(/ (* (pow (sin x) 2) (pow (sin eps) 2)) (* (pow (cos x) 2) (pow (cos eps) 2)))

prune1.6s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.4b

localize85.0ms

Local error

Found 4 expressions with local error:

1.9b
(- (/ (/ (sin x) (cos x)) (- 1 (* (* (/ (sin eps) (cos x)) (/ (sin eps) (cos x))) (* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))))) (/ (sin x) (cos x)))
0.3b
(* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))
0.3b
(* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))
0.3b
(* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))

rewrite69.0ms

Algorithm
rewrite-expression-head
Rules
67×add-sqr-sqrt
65×times-frac
52×*-un-lft-identity
46×add-cube-cbrt
40×add-exp-log
40×add-cbrt-cube
39×associate-*l*
39×associate-*r*
19×pow1
12×cbrt-unprod
12×prod-exp
12×div-exp
12×cbrt-undiv
6×difference-of-squares
6×add-log-exp
6×div-inv
3×pow-prod-up
3×*-commutative
3×associate-*r/
3×associate-*l/
3×pow-prod-down
3×pow-plus
3×frac-times
3×pow2
2×distribute-lft-out--
1×flip--
1×frac-sub
1×diff-log
1×flip3--
1×sub-neg
Counts
4 → 168
Calls
4 calls:
Slowest
34.0ms
(- (/ (/ (sin x) (cos x)) (- 1 (* (* (/ (sin eps) (cos x)) (/ (sin eps) (cos x))) (* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))))) (/ (sin x) (cos x)))
14.0ms
(* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))
10.0ms
(* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))
7.0ms
(* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))

series490.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
271.0ms
(- (/ (/ (sin x) (cos x)) (- 1 (* (* (/ (sin eps) (cos x)) (/ (sin eps) (cos x))) (* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))))) (/ (sin x) (cos x)))
79.0ms
(* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))
73.0ms
(* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))
66.0ms
(* (/ (sin x) (cos eps)) (/ (sin x) (cos eps)))

simplify12.3s

Counts
153 → 180
Calls
153 calls:
Slowest
1.1s
(* (* (* (/ (sin x) (cos eps)) (/ (sin x) (cos eps))) (/ (sin x) (cos eps))) (/ (* (* (sin x) (sin x)) (sin x)) (* (* (cos eps) (cos eps)) (cos eps))))
916.0ms
(* (* (* (/ (sin x) (cos eps)) (/ (sin x) (cos eps))) (/ (sin x) (cos eps))) (* (* (/ (sin x) (cos eps)) (/ (sin x) (cos eps))) (/ (sin x) (cos eps))))
868.0ms
(* (* (* (/ (sin x) (cos eps)) (/ (sin x) (cos eps))) (/ (sin x) (cos eps))) (* (* (/ (sin x) (cos eps)) (/ (sin x) (cos eps))) (/ (sin x) (cos eps))))
852.0ms
(* (* (* (/ (sin x) (cos eps)) (/ (sin x) (cos eps))) (/ (sin x) (cos eps))) (/ (* (* (sin x) (sin x)) (sin x)) (* (* (cos eps) (cos eps)) (cos eps))))
822.0ms
(* (* (* (/ (sin x) (cos eps)) (/ (sin x) (cos eps))) (/ (sin x) (cos eps))) (/ (* (* (sin x) (sin x)) (sin x)) (* (* (cos eps) (cos eps)) (cos eps))))

prune3.5s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.2b

regimes209.0ms

Accuracy

0% (0.3b remaining)

Error of 0.5b against oracle of 0.2b and baseline of 0.5b

bsearch18.0ms

end0.0ms

sample12.2s

Algorithm
intervals