Average Error: 31.5 → 0.2
Time: 16.1s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{1}{x} \cdot \left(\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

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

  3. Applied flip--31.7

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

  4. Applied associate-/l/31.7

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

  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 times-frac15.8

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

  8. Simplified15.7

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

  10. Applied *-un-lft-identity15.7

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

  11. Applied times-frac0.2

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

  12. Applied associate-*l*0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

Time bar (total: 15.3s)Debug log

sample116.0ms

Algorithm
intervals

simplify17.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
17.0ms
(/ (- 1 (cos x)) (* x x))

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 32.5b

localize11.0ms

Local error

Found 2 expressions with local error:

15.5b
(/ (- 1 (cos x)) (* x x))
2.0b
(- 1 (cos x))

rewrite5.0ms

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×log1p-expm1-u
2×flip3--
2×pow1
2×expm1-log1p-u
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 → 37
Calls
2 calls:
Slowest
4.0ms
(/ (- 1 (cos x)) (* x x))
1.0ms
(- 1 (cos x))

series62.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
33.0ms
(/ (- 1 (cos x)) (* x x))
28.0ms
(- 1 (cos x))

simplify796.0ms

Counts
19 → 43
Calls
19 calls:
Slowest
501.0ms
(/ (* (* (- 1 (cos x)) (- 1 (cos x))) (- 1 (cos x))) (* (* (* x x) (* x x)) (* x x)))
67.0ms
(- (+ (* 1/2 (pow x 2)) (* 1/720 (pow x 6))) (* 1/24 (pow x 4)))
65.0ms
(- (+ (* 1/720 (pow x 4)) 1/2) (* 1/24 (pow x 2)))
36.0ms
(- (log (- 1 (cos x))) (log (* x x)))
28.0ms
(/ (- 1 (cos x)) (pow x 2))

prune315.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 4 expressions with local error:

15.3b
(/ (* (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)))

rewrite30.0ms

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×log1p-expm1-u
4×associate-*r/
4×associate-*l*
4×div-exp
4×cbrt-undiv
4×expm1-log1p-u
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 → 84
Calls
4 calls:
Slowest
17.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
6.0ms
(* (* x x) (+ 1 (cos x)))
5.0ms
(* (sin x) (sin x))
2.0ms
(+ 1 (cos x))

series97.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
51.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
22.0ms
(* (* x x) (+ 1 (cos x)))
15.0ms
(* (sin x) (sin x))
9.0ms
(+ 1 (cos x))

simplify4.2s

Counts
51 → 96
Calls
51 calls:
Slowest
507.0ms
(/ (* (* (* (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)))))
413.0ms
(/ (* (* (* (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)))))
406.0ms
(- (+ (log (sin x)) (log (sin x))) (+ (log (* x x)) (log (+ 1 (cos x)))))
296.0ms
(/ (* (* (* (sin x) (sin x)) (* (sin x) (sin x))) (* (sin x) (sin x))) (* (* (* (* x x) (* x x)) (* x x)) (* (* (+ 1 (cos x)) (+ 1 (cos x))) (+ 1 (cos x)))))
292.0ms
(- (log (* (sin x) (sin x))) (+ (log (* x x)) (log (+ 1 (cos x)))))

prune884.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0b

localize10.0ms

Local error

Found 2 expressions with local error:

14.8b
(/ (sin x) (* x x))
0.2b
(* (/ (sin x) (* x x)) (tan (/ x 2)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
9×add-exp-log
9×add-cbrt-cube
7×add-cube-cbrt
7×associate-*l*
7×*-un-lft-identity
7×add-sqr-sqrt
6×times-frac
4×pow1
3×associate-/l*
3×associate-*r*
2×add-log-exp
2×div-inv
2×log1p-expm1-u
2×cbrt-unprod
2×tan-quot
2×prod-exp
2×div-exp
2×cbrt-undiv
2×expm1-log1p-u
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×frac-2neg
1×associate-/r*
1×clear-num
1×frac-times
Counts
2 → 49
Calls
2 calls:
Slowest
8.0ms
(* (/ (sin x) (* x x)) (tan (/ x 2)))
3.0ms
(/ (sin x) (* x x))

series86.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
65.0ms
(* (/ (sin x) (* x x)) (tan (/ x 2)))
21.0ms
(/ (sin x) (* x x))

simplify1.6s

Counts
34 → 55
Calls
34 calls:
Slowest
472.0ms
(* (* (* (/ (sin x) (* x x)) (/ (sin x) (* x x))) (/ (sin x) (* x x))) (* (* (tan (/ x 2)) (tan (/ x 2))) (tan (/ x 2))))
193.0ms
(/ (* (* (sin x) (sin x)) (sin x)) (* (* (* x x) (* x x)) (* x x)))
168.0ms
(/ (* (sin x) (sin (* 1/2 x))) (* (cos (* 1/2 x)) (pow x 2)))
155.0ms
(/ (* (sin x) (sin (* 1/2 x))) (* (cos (* 1/2 x)) (pow x 2)))
108.0ms
(+ (- (log (sin x)) (log (* x x))) (log (tan (/ x 2))))

prune567.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

localize17.0ms

Local error

Found 3 expressions with local error:

0.2b
(* (/ 1 x) (* (/ (sin x) x) (tan (/ x 2))))
0.2b
(* (/ (sin x) x) (tan (/ x 2)))
0.1b
(/ (sin x) x)

rewrite48.0ms

Algorithm
rewrite-expression-head
Rules
10×add-exp-log
10×add-cbrt-cube
10×pow1
8×associate-*l*
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×associate-*r/
6×tan-quot
6×frac-times
4×cbrt-unprod
4×prod-exp
4×associate-*l/
4×pow-prod-down
4×associate-*r*
3×add-log-exp
3×associate-/l*
3×div-inv
3×log1p-expm1-u
3×expm1-log1p-u
2×*-commutative
1×frac-2neg
1×clear-num
Counts
3 → 66
Calls
3 calls:
Slowest
31.0ms
(* (/ 1 x) (* (/ (sin x) x) (tan (/ x 2))))
12.0ms
(* (/ (sin x) x) (tan (/ x 2)))
4.0ms
(/ (sin x) x)

series220.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
97.0ms
(* (/ (sin x) x) (tan (/ x 2)))
91.0ms
(* (/ 1 x) (* (/ (sin x) x) (tan (/ x 2))))
31.0ms
(/ (sin x) x)

simplify2.1s

Counts
42 → 75
Calls
42 calls:
Slowest
332.0ms
(* (/ 1 x) (* (/ (sin x) x) (tan (/ x 2))))
262.0ms
(* (/ 1 x) (* (/ (sin x) x) (tan (/ x 2))))
206.0ms
(+ (log (/ 1 x)) (+ (log (/ (sin x) x)) (log (tan (/ x 2)))))
171.0ms
(+ (log (/ 1 x)) (log (* (/ (sin x) x) (tan (/ x 2)))))
167.0ms
(* (* (* (/ (sin x) x) (/ (sin x) x)) (/ (sin x) x)) (* (* (tan (/ x 2)) (tan (/ x 2))) (tan (/ x 2))))

prune521.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

regimes50.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch2.0ms

end0.0ms

sample3.6s

Algorithm
intervals