Average Error: 31.2 → 0.3
Time: 15.3s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\sin x}{\left(1 + \cos x\right) \cdot x} \cdot \frac{\sin x}{x}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.2

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

  3. Applied flip--31.3

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x}\]

  4. Applied associate-/l/31.3

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

  5. Simplified15.3

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

  7. Applied associate-*l*15.5

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

  9. Applied times-frac0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019022 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

Time bar (total: 14.7s)Debug log

sample94.0ms

Algorithm
intervals
Results
45.0ms59×body2560valid
16.0ms38×body1280valid
10.0ms32×body640valid
8.0ms37×body320valid
5.0ms74×body80valid
2.0ms16×body160valid

simplify23.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
23.0ms
(/ (- 1 (cos x)) (* x x))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.9b

localize21.0ms

Local error

Found 2 expressions with local error:

17.2b
(/ (- 1 (cos x)) (* x x))
1.2b
(- 1 (cos x))

rewrite34.0ms

Algorithm
rewrite-expression-head
Rules
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×add-sqr-sqrt
3×associate-/l*
3×times-frac
2×associate-/l/
2×add-log-exp
2×flip--
2×flip3--
2×pow1
1×div-inv
1×div-exp
1×div-sub
1×frac-2neg
1×sub-neg
1×associate-/r*
1×clear-num
1×cbrt-undiv
Counts
2 → 33
Calls
2 calls:
Slowest
32.0ms
(/ (- 1 (cos x)) (* x x))
1.0ms
(- 1 (cos x))

series57.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
32.0ms
(/ (- 1 (cos x)) (* x x))
25.0ms
(- 1 (cos x))

simplify588.0ms

Counts
19 → 39
Calls
19 calls:
Slowest
364.0ms
(/ (* (* (- 1 (cos x)) (- 1 (cos x))) (- 1 (cos x))) (* (* (* x x) (* x x)) (* x x)))
41.0ms
(- (+ (* 1/2 (pow x 2)) (* 1/720 (pow x 6))) (* 1/24 (pow x 4)))
36.0ms
(- (+ (* 1/720 (pow x 4)) 1/2) (* 1/24 (pow x 2)))
35.0ms
(- (log (- 1 (cos x))) (log (* x x)))
15.0ms
(/ (* (cbrt (- 1 (cos x))) (cbrt (- 1 (cos x)))) x)

prune231.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

17.1b
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
0.4b
(+ 1 (cos x))
0.2b
(* (sin x) (sin x))
0.1b
(* (* x x) (+ 1 (cos x)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
20×add-exp-log
20×add-cbrt-cube
11×pow1
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×cbrt-unprod
6×prod-exp
6×associate-*r*
4×add-log-exp
4×associate-*r/
4×associate-*l*
4×div-exp
4×cbrt-undiv
3×flip-+
3×flip3-+
2×sin-mult
2×sqr-sin
2×*-commutative
2×associate-/r/
2×pow-prod-down
1×associate-/l/
1×associate-/l*
1×div-inv
1×pow-prod-up
1×div-sub
1×pow-plus
1×times-frac
1×distribute-lft-in
1×frac-2neg
1×associate-/r*
1×clear-num
1×+-commutative
1×distribute-rgt-in
1×pow2
Counts
4 → 76
Calls
4 calls:
Slowest
8.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
4.0ms
(* (* x x) (+ 1 (cos x)))
3.0ms
(* (sin x) (sin x))
1.0ms
(+ 1 (cos x))

series102.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
41.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
30.0ms
(* (* x x) (+ 1 (cos x)))
22.0ms
(* (sin x) (sin x))
9.0ms
(+ 1 (cos x))

simplify4.4s

Counts
51 → 88
Calls
51 calls:
Slowest
1.0s
(* (* (* (* x x) (* x x)) (* x x)) (* (* (+ 1 (cos x)) (+ 1 (cos x))) (+ 1 (cos x))))
360.0ms
(/ (* (* (* (sin x) (sin x)) (sin x)) (* (* (sin x) (sin x)) (sin x))) (* (* (* (* x x) (+ 1 (cos x))) (* (* x x) (+ 1 (cos x)))) (* (* x x) (+ 1 (cos x)))))
329.0ms
(/ (* (* (* (sin x) (sin x)) (* (sin x) (sin x))) (* (sin x) (sin x))) (* (* (* (* x x) (+ 1 (cos x))) (* (* x x) (+ 1 (cos x)))) (* (* x x) (+ 1 (cos x)))))
284.0ms
(- (+ (log (sin x)) (log (sin x))) (+ (log (* x x)) (log (+ 1 (cos x)))))
280.0ms
(- (+ (log (sin x)) (log (sin x))) (log (* (* x x) (+ 1 (cos x)))))

prune764.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

localize14.0ms

Local error

Found 4 expressions with local error:

17.1b
(/ (* (sin x) (sin x)) (* x (* x (+ 1 (cos x)))))
0.4b
(+ 1 (cos x))
0.2b
(* (sin x) (sin x))
0.1b
(* x (+ 1 (cos x)))

rewrite22.0ms

Algorithm
rewrite-expression-head
Rules
11×add-exp-log
11×add-cbrt-cube
9×pow1
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×associate-*r/
6×associate-*r*
4×add-log-exp
3×flip-+
3×associate-*l*
3×flip3-+
2×sin-mult
2×sqr-sin
2×cbrt-unprod
2×*-commutative
2×associate-/r/
2×prod-exp
2×div-exp
2×cbrt-undiv
1×associate-/l/
1×associate-/l*
1×div-inv
1×pow-prod-up
1×pow-prod-down
1×div-sub
1×pow-plus
1×times-frac
1×distribute-lft-in
1×frac-2neg
1×associate-/r*
1×clear-num
1×+-commutative
1×distribute-rgt-in
1×pow2
Counts
4 → 68
Calls
4 calls:
Slowest
11.0ms
(/ (* (sin x) (sin x)) (* x (* x (+ 1 (cos x)))))
6.0ms
(* (sin x) (sin x))
2.0ms
(* x (+ 1 (cos x)))
1.0ms
(+ 1 (cos x))

series104.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
54.0ms
(/ (* (sin x) (sin x)) (* x (* x (+ 1 (cos x)))))
25.0ms
(* x (+ 1 (cos x)))
14.0ms
(* (sin x) (sin x))
11.0ms
(+ 1 (cos x))

simplify2.0s

Counts
44 → 80
Calls
44 calls:
Slowest
285.0ms
(- (+ (log (sin x)) (log (sin x))) (log (* x (* x (+ 1 (cos x))))))
263.0ms
(- (log (* (sin x) (sin x))) (log (* x (* x (+ 1 (cos x))))))
260.0ms
(/ (* (* (* (sin x) (sin x)) (sin x)) (* (* (sin x) (sin x)) (sin x))) (* (* (* x (* x (+ 1 (cos x)))) (* x (* x (+ 1 (cos x))))) (* x (* x (+ 1 (cos x))))))
229.0ms
(/ (pow (sin x) 2) (* (pow x 2) (+ (cos x) 1)))
189.0ms
(/ (pow (sin x) 2) (* (pow x 2) (+ (cos x) 1)))

prune619.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

0.4b
(+ 1 (cos x))
0.2b
(/ (sin x) (* x (+ 1 (cos x))))
0.1b
(/ (sin x) x)
0.1b
(* x (+ 1 (cos x)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×associate-/l*
6×add-exp-log
6×add-cbrt-cube
4×add-log-exp
4×associate-*r/
4×pow1
3×flip-+
3×associate-*r*
3×times-frac
3×flip3-+
2×div-inv
2×associate-/r/
2×frac-2neg
2×clear-num
1×*-commutative
1×div-exp
1×distribute-lft-in
1×associate-/r*
1×+-commutative
1×cbrt-undiv
1×distribute-rgt-in
Counts
4 → 59
Calls
4 calls:
Slowest
3.0ms
(/ (sin x) (* x (+ 1 (cos x))))
2.0ms
(/ (sin x) x)
1.0ms
(* x (+ 1 (cos x)))
1.0ms
(+ 1 (cos x))

series109.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
40.0ms
(/ (sin x) (* x (+ 1 (cos x))))
33.0ms
(* x (+ 1 (cos x)))
27.0ms
(/ (sin x) x)
8.0ms
(+ 1 (cos x))

simplify478.0ms

Counts
34 → 71
Calls
34 calls:
Slowest
86.0ms
(/ (* (* (sin x) (sin x)) (sin x)) (* (* (* x (+ 1 (cos x))) (* x (+ 1 (cos x)))) (* x (+ 1 (cos x)))))
68.0ms
(- (+ (* 2 x) (* 1/24 (pow x 5))) (* 1/2 (pow x 3)))
66.0ms
(+ (* 1 1) (- (* (cos x) (cos x)) (* 1 (cos x))))
54.0ms
(+ (* 1 1) (- (* (cos x) (cos x)) (* 1 (cos x))))
40.0ms
(- (+ (* 1/24 (pow x 4)) 2) (* 1/2 (pow x 2)))

prune604.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

regimes69.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch1.0ms

end0.0ms

sample4.4s

Algorithm
intervals
Results
1.7s1596×body2560valid
826.0ms1253×body1280valid
795.0ms1458×body640valid
570.0ms2280×body80valid
310.0ms936×body320valid
98.0ms477×body160valid