Average Error: 30.5 → 0.1
Time: 36.2s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\frac{\tan \left(\frac{x}{2}\right)}{x}}{x}}{\frac{1}{\sin x}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.5

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

  3. Applied flip--30.6

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

  4. Applied associate-/l/30.6

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

  5. Simplified15.2

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

  6. Taylor expanded around -inf 15.2

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

  7. Simplified15.4

    \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{\frac{x \cdot x}{\sin x}}}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity15.4

    \[\leadsto \frac{\tan \left(\frac{x}{2}\right)}{\frac{x \cdot x}{\color{blue}{1 \cdot \sin x}}}\]

  10. Applied times-frac0.5

    \[\leadsto \frac{\tan \left(\frac{x}{2}\right)}{\color{blue}{\frac{x}{1} \cdot \frac{x}{\sin x}}}\]

  11. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{\tan \left(\frac{x}{2}\right)}{\frac{x}{1}}}{\frac{x}{\sin x}}}\]
  12. Using strategy rm

  13. Applied div-inv0.2

    \[\leadsto \frac{\frac{\tan \left(\frac{x}{2}\right)}{\frac{x}{1}}}{\color{blue}{x \cdot \frac{1}{\sin x}}}\]

  14. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{\tan \left(\frac{x}{2}\right)}{\frac{x}{1}}}{x}}{\frac{1}{\sin x}}}\]

  15. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{\tan \left(\frac{x}{2}\right)}{x}}{x}}{\frac{1}{\sin x}}\]

Reproduce

herbie shell --seed 323378853 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

Time bar (total: 34.8s)Debug log

sample200.0ms

Algorithm
intervals
Results
74.0ms50×body2560valid
42.0ms42×body1280valid
39.0ms54×body640valid
15.0ms28×body320valid
11.0ms62×body80valid
7.0ms20×body160valid

simplify33.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
33.0ms
(/ (- 1 (cos x)) (* x x))

prune31.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 31.3b

localize34.0ms

Local error

Found 2 expressions with local error:

15.2b
(/ (- 1 (cos x)) (* x x))
1.5b
(- 1 (cos x))

rewrite12.0ms

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
9.0ms
(/ (- 1 (cos x)) (* x x))
2.0ms
(- 1 (cos x))

series98.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
61.0ms
(/ (- 1 (cos x)) (* x x))
36.0ms
(- 1 (cos x))

simplify934.0ms

Counts
19 → 39
Calls
19 calls:
Slowest
527.0ms
(/ (* (* (- 1 (cos x)) (- 1 (cos x))) (- 1 (cos x))) (* (* (* x x) (* x x)) (* x x)))
82.0ms
(- (+ (* 1/2 (pow x 2)) (* 1/720 (pow x 6))) (* 1/24 (pow x 4)))
79.0ms
(- (+ (* 1/720 (pow x 4)) 1/2) (* 1/24 (pow x 2)))
49.0ms
(- (log (- 1 (cos x))) (log (* x x)))
35.0ms
(/ (- 1 (cos x)) (pow x 2))

prune535.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize38.0ms

Local error

Found 4 expressions with local error:

15.0b
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
0.5b
(+ 1 (cos x))
0.2b
(* (sin x) (sin x))
0.1b
(* (* x x) (+ 1 (cos x)))

rewrite40.0ms

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
20.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
9.0ms
(* (* x x) (+ 1 (cos x)))
8.0ms
(* (sin x) (sin x))
2.0ms
(+ 1 (cos x))

series174.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
81.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
41.0ms
(* (* x x) (+ 1 (cos x)))
31.0ms
(* (sin x) (sin x))
21.0ms
(+ 1 (cos x))

simplify7.1s

Counts
51 → 88
Calls
51 calls:
Slowest
1.4s
(* (* (* (* x x) (* x x)) (* x x)) (* (* (+ 1 (cos x)) (+ 1 (cos x))) (+ 1 (cos x))))
619.0ms
(/ (* (* (* (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)))))
552.0ms
(/ (* (* (* (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)))))
499.0ms
(- (+ (log (sin x)) (log (sin x))) (log (* (* x x) (+ 1 (cos x)))))
485.0ms
(- (+ (log (sin x)) (log (sin x))) (+ (log (* x x)) (log (+ 1 (cos x)))))

prune1.4s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

localize31.0ms

Local error

Found 2 expressions with local error:

14.2b
(/ (* x x) (sin x))
0.5b
(/ (tan (/ x 2)) (/ (* x x) (sin x)))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
36×times-frac
20×add-cube-cbrt
20×*-un-lft-identity
20×add-sqr-sqrt
10×associate-/r*
9×add-exp-log
9×add-cbrt-cube
6×div-inv
4×associate-/l*
4×div-exp
4×cbrt-undiv
2×add-log-exp
2×frac-2neg
2×clear-num
2×pow1
1×associate-/l/
1×associate-/r/
1×tan-quot
Counts
2 → 66
Calls
2 calls:
Slowest
12.0ms
(/ (tan (/ x 2)) (/ (* x x) (sin x)))
7.0ms
(/ (* x x) (sin x))

series171.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
118.0ms
(/ (tan (/ x 2)) (/ (* x x) (sin x)))
53.0ms
(/ (* x x) (sin x))

simplify3.7s

Counts
74 → 72
Calls
74 calls:
Slowest
1.1s
(/ (* (* (tan (/ x 2)) (tan (/ x 2))) (tan (/ x 2))) (* (* (/ (* x x) (sin x)) (/ (* x x) (sin x))) (/ (* x x) (sin x))))
362.0ms
(/ (* (* (* x x) (* x x)) (* x x)) (* (* (sin x) (sin x)) (sin x)))
260.0ms
(/ (* (sin x) (sin (* 1/2 x))) (* (cos (* 1/2 x)) (pow x 2)))
241.0ms
(/ (* (sin x) (sin (* 1/2 x))) (* (cos (* 1/2 x)) (pow x 2)))
152.0ms
(- (log (tan (/ x 2))) (- (log (* x x)) (log (sin x))))

prune1.1s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0b

localize29.0ms

Local error

Found 3 expressions with local error:

0.1b
(/ (tan (/ x 2)) (/ x 1))
0.1b
(/ x (sin x))
0.1b
(/ (/ (tan (/ x 2)) (/ x 1)) (/ x (sin x)))

rewrite32.0ms

Algorithm
rewrite-expression-head
Rules
140×times-frac
71×add-cube-cbrt
71×*-un-lft-identity
71×add-sqr-sqrt
45×div-inv
20×associate-/l*
11×associate-/r*
10×add-exp-log
10×add-cbrt-cube
7×associate-/r/
4×div-exp
4×cbrt-undiv
3×add-log-exp
3×frac-2neg
3×clear-num
3×pow1
2×associate-/l/
1×tan-quot
Counts
3 → 151
Calls
3 calls:
Slowest
15.0ms
(/ (/ (tan (/ x 2)) (/ x 1)) (/ x (sin x)))
11.0ms
(/ (tan (/ x 2)) (/ x 1))
1.0ms
(/ x (sin x))

series241.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
117.0ms
(/ (/ (tan (/ x 2)) (/ x 1)) (/ x (sin x)))
76.0ms
(/ (tan (/ x 2)) (/ x 1))
48.0ms
(/ x (sin x))

simplify9.2s

Counts
207 → 160
Calls
207 calls:
Slowest
990.0ms
(/ (* (* (/ (tan (/ x 2)) (/ x 1)) (/ (tan (/ x 2)) (/ x 1))) (/ (tan (/ x 2)) (/ x 1))) (* (* (/ x (sin x)) (/ x (sin x))) (/ x (sin x))))
376.0ms
(/ (/ (* (cbrt (tan (/ x 2))) (cbrt (tan (/ x 2)))) (* (cbrt (/ x 1)) (cbrt (/ x 1)))) (* (cbrt (/ x (sin x))) (cbrt (/ x (sin x)))))
364.0ms
(- (- (log (tan (/ x 2))) (log (/ x 1))) (log (/ x (sin x))))
318.0ms
(/ (* (* (tan (/ x 2)) (tan (/ x 2))) (tan (/ x 2))) (* (* (/ x 1) (/ x 1)) (/ x 1)))
312.0ms
(/ (/ (* (cbrt (tan (/ x 2))) (cbrt (tan (/ x 2)))) (* (cbrt (/ x 1)) (cbrt (/ x 1)))) 1)

prune2.5s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0b

regimes59.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch3.0ms

end0.0ms

sample7.1s

Algorithm
intervals
Results
3.1s1564×body2560valid
1.4s1241×body1280valid
1.2s1539×body640valid
634.0ms938×body320valid
486.0ms2241×body80valid
179.0ms477×body160valid