Average Error: 0.3 → 0.5
Time: 17.7s
Precision: 64
Internal Precision: 128
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\left(1 - e^{\log \left(\tan x \cdot \tan x\right)}\right) \cdot \frac{1 - \tan x \cdot \tan x}{1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.3

    \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
  2. Using strategy rm

  3. Applied flip-+0.4

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

  4. Applied associate-/r/0.4

    \[\leadsto \color{blue}{\frac{1 - \tan x \cdot \tan x}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)} \cdot \left(1 - \tan x \cdot \tan x\right)}\]
  5. Using strategy rm

  6. Applied add-exp-log0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019022 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))

Details

Time bar (total: 16.7s)Debug log

sample263.0ms

Algorithm
intervals
Results
106.0ms45×body1280valid
28.0ms143×body80valid
28.0ms31×body640valid
16.0ms25×body320valid
5.0ms12×body160valid

simplify67.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
67.0ms
(/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x))))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
16×add-cube-cbrt
16×*-un-lft-identity
16×add-sqr-sqrt
14×pow1
10×add-exp-log
10×add-cbrt-cube
9×times-frac
8×tan-quot
6×associate-*l*
6×associate-*r*
4×add-log-exp
3×associate-/l*
3×associate-/r*
2×associate-/l/
2×flip--
2×cbrt-unprod
2×pow-prod-up
2×*-commutative
2×associate-/r/
2×associate-*r/
2×prod-exp
2×associate-*l/
2×pow-prod-down
2×flip3--
2×pow-plus
2×frac-times
2×pow2
1×div-inv
1×flip-+
1×div-exp
1×div-sub
1×flip3-+
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
4 → 89
Calls
4 calls:
Slowest
5.0ms
(* (tan x) (tan x))
4.0ms
(/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x))))
4.0ms
(* (tan x) (tan x))
1.0ms
(- 1 (* (tan x) (tan x)))

series222.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
100.0ms
(/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x))))
52.0ms
(* (tan x) (tan x))
46.0ms
(* (tan x) (tan x))
23.0ms
(- 1 (* (tan x) (tan x)))

simplify1.7s

Counts
72 → 101
Calls
72 calls:
Slowest
419.0ms
(/ (* (* (- 1 (* (tan x) (tan x))) (- 1 (* (tan x) (tan x)))) (- 1 (* (tan x) (tan x)))) (* (* (+ 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))) (+ 1 (* (tan x) (tan x)))))
213.0ms
(+ (* 1 1) (- (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* 1 (* (tan x) (tan x)))))
165.0ms
(/ (- 1 (/ (pow (sin x) 2) (pow (cos x) 2))) (+ (/ (pow (sin x) 2) (pow (cos x) 2)) 1))
159.0ms
(/ (- 1 (/ (pow (sin x) 2) (pow (cos x) 2))) (+ (/ (pow (sin x) 2) (pow (cos x) 2)) 1))
47.0ms
(- (pow 1 3) (pow (* (tan x) (tan x)) 3))

prune902.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
15×pow1
9×add-cube-cbrt
9×add-exp-log
9×*-un-lft-identity
9×add-sqr-sqrt
8×tan-quot
8×add-cbrt-cube
6×associate-*l*
6×associate-*r*
4×add-log-exp
3×associate-/r*
2×cbrt-unprod
2×pow-prod-up
2×*-commutative
2×associate-/r/
2×associate-*r/
2×prod-exp
2×associate-*l/
2×pow-prod-down
2×pow-plus
2×frac-times
2×pow2
1×inv-pow
1×flip--
1×pow-flip
1×div-inv
1×flip-+
1×flip3--
1×flip3-+
1×frac-2neg
1×sub-neg
1×clear-num
1×rec-exp
Counts
4 → 75
Calls
4 calls:
Slowest
7.0ms
(* (tan x) (tan x))
7.0ms
(* (tan x) (tan x))
2.0ms
(- 1 (* (tan x) (tan x)))
2.0ms
(/ 1 (+ 1 (* (tan x) (tan x))))

series123.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
39.0ms
(/ 1 (+ 1 (* (tan x) (tan x))))
31.0ms
(* (tan x) (tan x))
28.0ms
(* (tan x) (tan x))
24.0ms
(- 1 (* (tan x) (tan x)))

simplify593.0ms

Counts
48 → 87
Calls
48 calls:
Slowest
256.0ms
(+ (* 1 1) (- (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* 1 (* (tan x) (tan x)))))
46.0ms
(- 1 (+ (pow x 2) (* 2/3 (pow x 4))))
41.0ms
(- 1 (/ (pow (sin x) 2) (pow (cos x) 2)))
30.0ms
(- 1 (/ (pow (sin x) 2) (pow (cos x) 2)))
24.0ms
(/ 1 (+ (/ (pow (sin x) 2) (pow (cos x) 2)) 1))

prune722.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
0.3b
(* (tan x) (tan x))
0.3b
(* (tan x) (tan x))
0.3b
(* (tan x) (tan x))

rewrite49.0ms

Algorithm
rewrite-expression-head
Rules
76×pow1
44×tan-quot
26×pow-prod-up
22×add-exp-log
22×add-cbrt-cube
21×pow-prod-down
20×frac-times
15×pow-plus
14×associate-*r/
14×associate-*l/
12×pow2
11×cbrt-unprod
11×prod-exp
10×add-cube-cbrt
10×associate-*l*
10×associate-*r*
10×*-un-lft-identity
10×add-sqr-sqrt
4×add-log-exp
4×*-commutative
Counts
4 → 127
Calls
4 calls:
Slowest
31.0ms
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
8.0ms
(* (tan x) (tan x))
4.0ms
(* (tan x) (tan x))
4.0ms
(* (tan x) (tan x))

series151.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
45.0ms
(* (tan x) (tan x))
41.0ms
(* (tan x) (tan x))
39.0ms
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
26.0ms
(* (tan x) (tan x))

simplify970.0ms

Counts
132 → 139
Calls
132 calls:
Slowest
133.0ms
(* (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))) (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))))
92.0ms
(* (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))) (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))))
91.0ms
(* (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))) (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))))
78.0ms
(* (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))) (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))))
30.0ms
(* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x)))

prune1.5s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

17.4b
(log (* (tan x) (tan x)))
1.5b
(exp (log (* (tan x) (tan x))))
0.3b
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
0.3b
(* (tan x) (tan x))

rewrite62.0ms

Algorithm
rewrite-expression-head
Rules
78×pow1
44×tan-quot
26×pow-prod-up
21×add-exp-log
21×pow-prod-down
20×frac-times
18×add-cbrt-cube
15×pow-plus
14×associate-*r/
14×associate-*l/
12×pow2
10×log-pow
10×prod-exp
9×cbrt-unprod
8×exp-prod
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×log-div
4×add-log-exp
4×associate-*l*
4×associate-*r*
3×exp-diff
2×*-commutative
2×rem-log-exp
2×log-prod
1×rem-exp-log
1×exp-sum
Counts
4 → 119
Calls
4 calls:
Slowest
31.0ms
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
13.0ms
(exp (log (* (tan x) (tan x))))
7.0ms
(* (tan x) (tan x))
3.0ms
(log (* (tan x) (tan x)))

series261.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
119.0ms
(log (* (tan x) (tan x)))
55.0ms
(* (* (tan x) (tan x)) (* (tan x) (tan x)))
46.0ms
(exp (log (* (tan x) (tan x))))
41.0ms
(* (tan x) (tan x))

simplify959.0ms

Counts
123 → 131
Calls
123 calls:
Slowest
94.0ms
(* (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))) (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))))
88.0ms
(* (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))) (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))))
85.0ms
(* (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))) (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))))
79.0ms
(* (* (* (* (tan x) (tan x)) (* (tan x) (tan x))) (* (tan x) (tan x))) (* (* (* (tan x) (tan x)) (tan x)) (* (* (tan x) (tan x)) (tan x))))
40.0ms
(log (/ (pow (sin x) 2) (pow (cos x) 2)))

prune1.6s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.0b

regimes423.0ms

Accuracy

0% (0.4b remaining)

Error of 0.5b against oracle of 0.1b and baseline of 0.5b

bsearch2.0ms

end0.0ms

sample6.1s

Algorithm
intervals
Results
3.2s1523×body1280valid
1.2s1167×body640valid
999.0ms4354×body80valid
399.0ms632×body320valid
179.0ms324×body160valid