Average Error: 31.2 → 0.3
Time: 18.4s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\sin x}{x} \cdot \frac{\sin x}{(\left(\cos x\right) \cdot x + x)_*}\]

Error

Bits error versus x

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. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

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

Details

Time bar (total: 17.9s)Debug log

sample113.0ms

Algorithm
intervals
Results
53.0ms59×body2560valid
20.0ms38×body1280valid
11.0ms32×body640valid
9.0ms37×body320valid
7.0ms74×body80valid
3.0ms16×body160valid

simplify28.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
27.0ms
(/ (- 1 (cos x)) (* x x))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.9b

localize38.0ms

Local error

Found 2 expressions with local error:

17.2b
(/ (- 1 (cos x)) (* x x))
1.2b
(- 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×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
9.0ms
(/ (- 1 (cos x)) (* x x))
1.0ms
(- 1 (cos x))

series65.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
48.0ms
(/ (- 1 (cos x)) (* x x))
17.0ms
(- 1 (cos x))

simplify664.0ms

Counts
19 → 43
Calls
19 calls:
Slowest
334.0ms
(/ (* (* (- 1 (cos x)) (- 1 (cos x))) (- 1 (cos x))) (* (* (* x x) (* x x)) (* x x)))
75.0ms
(- (+ (* 1/720 (pow x 4)) 1/2) (* 1/24 (pow x 2)))
70.0ms
(- (+ (* 1/2 (pow x 2)) (* 1/720 (pow x 6))) (* 1/24 (pow x 4)))
31.0ms
(- (log (- 1 (cos x))) (log (* x x)))
25.0ms
(/ (- 1 (cos x)) (pow x 2))

prune292.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 4 expressions with local error:

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)))

rewrite32.0ms

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
15.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
7.0ms
(* (* x x) (+ 1 (cos x)))
7.0ms
(* (sin x) (sin x))
1.0ms
(+ 1 (cos x))

series133.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
65.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
29.0ms
(* (* x x) (+ 1 (cos x)))
27.0ms
(* (sin x) (sin x))
13.0ms
(+ 1 (cos x))

simplify4.1s

Counts
51 → 96
Calls
51 calls:
Slowest
536.0ms
(/ (* (* (* (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)))))
370.0ms
(/ (* (* (* (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)))))
338.0ms
(- (+ (log (sin x)) (log (sin x))) (+ (log (* x x)) (log (+ 1 (cos x)))))
297.0ms
(/ (* (* (* (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)))))
296.0ms
(/ (pow (sin x) 2) (* (pow x 2) (+ (cos x) 1)))

prune795.0ms

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)))

rewrite21.0ms

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
4×log1p-expm1-u
4×expm1-log1p-u
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 → 76
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))

series113.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
55.0ms
(/ (* (sin x) (sin x)) (* x (* x (+ 1 (cos x)))))
35.0ms
(* x (+ 1 (cos x)))
14.0ms
(* (sin x) (sin x))
9.0ms
(+ 1 (cos x))

simplify2.4s

Counts
44 → 88
Calls
44 calls:
Slowest
411.0ms
(- (+ (log (sin x)) (log (sin x))) (log (* x (* x (+ 1 (cos x))))))
294.0ms
(- (log (* (sin x) (sin x))) (log (* x (* x (+ 1 (cos x))))))
270.0ms
(/ (* (* (* (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))))))
265.0ms
(/ (* (* (* (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))))))
226.0ms
(/ (pow (sin x) 2) (* (pow x 2) (+ (cos x) 1)))

prune684.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0b

localize11.0ms

Local error

Found 4 expressions with local error:

0.4b
(fma (cos x) x x)
0.2b
(/ (sin x) (fma (cos x) x x))
0.1b
(/ (sin x) x)
0.1b
(* (/ (sin x) x) (/ (sin x) (fma (cos x) x x)))

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
21×add-cube-cbrt
21×*-un-lft-identity
21×add-sqr-sqrt
18×times-frac
13×associate-*r*
11×add-exp-log
11×add-cbrt-cube
6×associate-/l*
6×pow1
4×add-log-exp
4×div-inv
4×log1p-expm1-u
4×associate-*l*
4×expm1-log1p-u
3×associate-/r*
2×cbrt-unprod
2×prod-exp
2×div-exp
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×fma-udef
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×frac-times
Counts
4 → 89
Calls
4 calls:
Slowest
12.0ms
(* (/ (sin x) x) (/ (sin x) (fma (cos x) x x)))
3.0ms
(/ (sin x) (fma (cos x) x x))
2.0ms
(/ (sin x) x)
0.0ms
(fma (cos x) x x)

series186.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
77.0ms
(* (/ (sin x) x) (/ (sin x) (fma (cos x) x x)))
48.0ms
(/ (sin x) (fma (cos x) x x))
39.0ms
(fma (cos x) x x)
22.0ms
(/ (sin x) x)

simplify2.6s

Counts
64 → 101
Calls
64 calls:
Slowest
546.0ms
(* (* (* (/ (sin x) x) (/ (sin x) x)) (/ (sin x) x)) (* (* (/ (sin x) (fma (cos x) x x)) (/ (sin x) (fma (cos x) x x))) (/ (sin x) (fma (cos x) x x))))
298.0ms
(+ (log (/ (sin x) x)) (- (log (sin x)) (log (fma (cos x) x x))))
195.0ms
(/ (pow (sin x) 2) (* (pow x 2) (+ (cos x) 1)))
181.0ms
(/ (pow (sin x) 2) (* (pow x 2) (+ (cos x) 1)))
156.0ms
(/ (* (* (sin x) (sin x)) (sin x)) (* (* (fma (cos x) x x) (fma (cos x) x x)) (fma (cos x) x x)))

prune752.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

regimes91.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms

end0.0ms

sample4.7s

Algorithm
intervals
Results
1.6s1596×body2560valid
1.2s1253×body1280valid
1.1s1458×body640valid
382.0ms936×body320valid
217.0ms2280×body80valid
98.0ms477×body160valid