Average Error: 36.8 → 0.6
Time: 58.5s
Precision: 64
Internal Precision: 128
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right) + \left(\frac{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin x}{\cos x}}{1 - \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right)} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x} + 1\right)}{1 - \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right)}\right)\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target14.9
Herbie0.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.8

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

  3. Applied tan-sum21.9

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

  5. Applied flip--21.9

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

  6. Applied associate-/r/21.9

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

  7. Taylor expanded around -inf 22.0

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

  8. Simplified0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 323378853 
(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: 55.4s)Debug log

sample635.0ms

Algorithm
intervals
Results
370.0ms116×body1280valid
93.0ms19×body2560valid
88.0ms46×body640valid
50.0ms34×body320valid
13.0ms32×body80valid
8.0ms9×body160valid

simplify8.0ms

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

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 38.3b

localize62.0ms

Local error

Found 2 expressions with local error:

3.6b
(tan (+ x eps))
1.3b
(- (tan (+ x eps)) (tan x))

rewrite19.0ms

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
15.0ms
(- (tan (+ x eps)) (tan x))
3.0ms
(tan (+ x eps))

series241.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
155.0ms
(- (tan (+ x eps)) (tan x))
86.0ms
(tan (+ x eps))

simplify817.0ms

Counts
15 → 31
Calls
15 calls:
Slowest
274.0ms
(- (* (+ (tan x) (tan eps)) (cos x)) (* (- 1 (* (tan x) (tan eps))) (sin x)))
183.0ms
(* (- 1 (* (tan x) (tan eps))) (cos x))
182.0ms
(+ (* x (pow eps 2)) (+ eps (* (pow x 2) eps)))
62.0ms
(+ x (+ (* 1/3 (pow x 3)) eps))
28.0ms
(/ (exp (tan (+ x eps))) (exp (tan x)))

prune516.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 15.8b

localize46.0ms

Local error

Found 4 expressions with local error:

2.9b
(- (/ (+ (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
(/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps))))

rewrite117.0ms

Algorithm
rewrite-expression-head
Rules
27×*-un-lft-identity
20×add-sqr-sqrt
15×add-cube-cbrt
13×times-frac
9×tan-quot
8×add-log-exp
8×add-exp-log
8×add-cbrt-cube
6×pow1
5×distribute-lft-out
4×associate-/l*
3×associate-/l/
3×associate-*l*
3×associate-*r*
3×associate-/r*
2×difference-of-squares
2×flip--
2×flip-+
2×associate-/r/
2×frac-add
2×flip3--
2×flip3-+
1×distribute-lft-out--
1×div-inv
1×cbrt-unprod
1×frac-sub
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×div-exp
1×diff-log
1×frac-2neg
1×sub-neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
1×frac-times
Counts
4 → 85
Calls
4 calls:
Slowest
67.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x))
29.0ms
(/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps))))
10.0ms
(* (tan x) (tan eps))
8.0ms
(+ (tan x) (tan eps))

series1.3s

Counts
4 → 12
Calls
4 calls:
Slowest
625.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x))
373.0ms
(/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps))))
167.0ms
(+ (tan x) (tan eps))
163.0ms
(* (tan x) (tan eps))

simplify3.9s

Counts
72 → 97
Calls
72 calls:
Slowest
436.0ms
(+ (* 1/3 (* (pow x 3) eps)) (+ (* 1/3 (* x (pow eps 3))) (* x eps)))
417.0ms
(/ (* (* (+ (tan x) (tan eps)) (+ (tan x) (tan eps))) (+ (tan x) (tan eps))) (* (* (- 1 (* (tan x) (tan eps))) (- 1 (* (tan x) (tan eps)))) (- 1 (* (tan x) (tan eps)))))
348.0ms
(- (+ (/ (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)))
345.0ms
(- (+ (/ (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)))
275.0ms
(- (* (+ (tan x) (tan eps)) (cos x)) (* (- 1 (* (tan x) (tan eps))) (sin x)))

prune2.2s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 15.7b

localize53.0ms

Local error

Found 4 expressions with local error:

2.9b
(- (/ (+ (tan x) (tan eps)) (- 1 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))
0.2b
(/ (* (tan x) (sin eps)) (cos eps))
0.2b
(* (tan x) (sin eps))
0.1b
(+ (tan x) (tan eps))

rewrite113.0ms

Algorithm
rewrite-expression-head
Rules
13×add-sqr-sqrt
12×*-un-lft-identity
11×add-exp-log
11×add-cbrt-cube
8×add-log-exp
8×add-cube-cbrt
6×pow1
5×tan-quot
4×times-frac
3×associate-*l*
3×associate-*r*
3×associate-/r*
2×difference-of-squares
2×cbrt-unprod
2×prod-exp
2×associate-*l/
2×div-exp
2×cbrt-undiv
1×associate-/l/
1×distribute-lft-out--
1×associate-/l*
1×distribute-lft-out
1×flip--
1×div-inv
1×flip-+
1×frac-sub
1×*-commutative
1×frac-add
1×pow-prod-down
1×diff-log
1×flip3--
1×flip3-+
1×frac-2neg
1×sub-neg
1×sum-log
1×clear-num
1×+-commutative
Counts
4 → 69
Calls
4 calls:
Slowest
78.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))
16.0ms
(/ (* (tan x) (sin eps)) (cos eps))
9.0ms
(* (tan x) (sin eps))
8.0ms
(+ (tan x) (tan eps))

series1.1s

Counts
4 → 12
Calls
4 calls:
Slowest
635.0ms
(- (/ (+ (tan x) (tan eps)) (- 1 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))
171.0ms
(/ (* (tan x) (sin eps)) (cos eps))
166.0ms
(+ (tan x) (tan eps))
120.0ms
(* (tan x) (sin eps))

simplify4.7s

Counts
48 → 81
Calls
48 calls:
Slowest
680.0ms
(/ (* (* (* (tan x) (sin eps)) (* (tan x) (sin eps))) (* (tan x) (sin eps))) (* (* (cos eps) (cos eps)) (cos eps)))
658.0ms
(/ (* (* (* (tan x) (tan x)) (tan x)) (* (* (sin eps) (sin eps)) (sin eps))) (* (* (cos eps) (cos eps)) (cos eps)))
443.0ms
(+ (* 1/3 (* (pow x 3) eps)) (+ (* 1/3 (* x (pow eps 3))) (* x eps)))
400.0ms
(- (+ (* 1/3 (* (pow x 3) eps)) (* x eps)) (* 1/6 (* x (pow eps 3))))
391.0ms
(* (- 1 (/ (* (tan x) (sin eps)) (cos eps))) (cos x))

prune1.8s

Pruning

16 alts after pruning (15 fresh and 1 done)

Merged error: 15.7b

localize51.0ms

Local error

Found 4 expressions with local error:

2.9b
(- (* (/ (+ (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))

rewrite87.0ms

Algorithm
rewrite-expression-head
Rules
45×tan-quot
35×pow1
21×frac-times
20×add-exp-log
20×add-cbrt-cube
15×associate-*r/
15×pow-prod-down
14×associate-*l/
10×cbrt-unprod
10×prod-exp
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×associate-*l*
7×associate-*r*
6×add-log-exp
5×frac-sub
4×pow-prod-up
3×*-commutative
2×flip-+
2×associate--l+
2×pow-plus
2×flip3-+
1×flip--
1×diff-log
1×flip3--
1×distribute-lft-in
1×sub-neg
1×distribute-rgt-in
1×pow2
Counts
4 → 103
Calls
4 calls:
Slowest
39.0ms
(* (* (tan x) (tan eps)) (* (tan x) (tan eps)))
25.0ms
(- (* (/ (+ (tan x) (tan eps)) (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))) (+ 1 (* (tan x) (tan eps)))) (tan x))
9.0ms
(* (tan x) (tan eps))
9.0ms
(* (tan x) (tan eps))

series2.0s

Counts
4 → 12
Calls
4 calls:
Slowest
1.5s
(- (* (/ (+ (tan x) (tan eps)) (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))) (+ 1 (* (tan x) (tan eps)))) (tan x))
191.0ms
(* (* (tan x) (tan eps)) (* (tan x) (tan eps)))
167.0ms
(* (tan x) (tan eps))
166.0ms
(* (tan x) (tan eps))

simplify11.3s

Counts
97 → 115
Calls
97 calls:
Slowest
866.0ms
(* (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan eps) (tan eps)) (tan eps))) (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))))
813.0ms
(* (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))) (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan eps) (tan eps)) (tan eps))))
791.0ms
(* (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))) (* (* (* (tan x) (tan eps)) (* (tan x) (tan eps))) (* (tan x) (tan eps))))
590.0ms
(- (* (* (+ (tan x) (tan eps)) (+ 1 (* (tan x) (tan eps)))) (cos x)) (* (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps)))) (sin x)))
532.0ms
(- (* (* (+ (tan x) (tan eps)) (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))) (cos x)) (* (* (- (* 1 1) (* (* (tan x) (tan eps)) (* (tan x) (tan eps)))) (- 1 (* (tan x) (tan eps)))) (sin x)))

prune2.3s

Pruning

14 alts after pruning (13 fresh and 1 done)

Merged error: 0.2b

regimes359.0ms

Accuracy

0% (0.3b remaining)

Error of 0.6b against oracle of 0.3b and baseline of 0.6b

bsearch8.0ms

end0.0ms

sample21.6s

Algorithm
intervals
Results
11.6s3299×body1280valid
4.0s1865×body640valid
3.9s670×body2560valid
1.1s750×body320valid
509.0ms1043×body80valid
338.0ms373×body160valid