Average Error: 30.9 → 0.2
Time: 31.7s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03247091185753463 \lor \neg \left(x \le 0.03064339360211378\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;(\frac{1}{720} \cdot \left({x}^{4}\right) + \left((\left(\frac{-1}{24} \cdot x\right) \cdot x + \frac{1}{2})_*\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03247091185753463 or 0.03064339360211378 < x

    1. Initial program 1.1

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

    3. Applied associate-/r*0.5

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

    if -0.03247091185753463 < x < 0.03064339360211378

    1. Initial program 61.3

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

    3. Applied flip--61.3

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

    4. Applied associate-/l/61.3

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

    5. Simplified29.9

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

    6. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    7. Simplified0.0

      \[\leadsto \color{blue}{(\frac{1}{720} \cdot \left({x}^{4}\right) + \left((\left(x \cdot \frac{-1}{24}\right) \cdot x + \frac{1}{2})_*\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03247091185753463 \lor \neg \left(x \le 0.03064339360211378\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;(\frac{1}{720} \cdot \left({x}^{4}\right) + \left((\left(\frac{-1}{24} \cdot x\right) \cdot x + \frac{1}{2})_*\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

Time bar (total: 30.9s)Debug log

sample174.0ms

Algorithm
intervals

simplify27.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

27.0ms
(/ (- 1 (cos x)) (* x x))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 27.6b

localize22.0ms

Local error

Found 2 expressions with local error:

15.2b
(/ (- 1 (cos x)) (* x x))
2.6b
(- 1 (cos x))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
2 → 37
Calls

2 calls. Slowest were:

7.0ms
(/ (- 1 (cos x)) (* x x))
2.0ms
(- 1 (cos x))

series73.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

49.0ms
(/ (- 1 (cos x)) (* x x))
24.0ms
(- 1 (cos x))

simplify687.0ms

Counts
19 → 43
Calls

19 calls. Slowest were:

394.0ms
(/ (* (* (- 1 (cos x)) (- 1 (cos x))) (- 1 (cos x))) (* (* (* x x) (* x x)) (* x x)))
88.0ms
(- (+ (* 1/2 (pow x 2)) (* 1/720 (pow x 6))) (* 1/24 (pow x 4)))
71.0ms
(- (+ (* 1/720 (pow x 4)) 1/2) (* 1/24 (pow x 2)))

prune376.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize29.0ms

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)))

rewrite38.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

20.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
7.0ms
(* (* x x) (+ 1 (cos x)))
7.0ms
(* (sin x) (sin x))

series122.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

48.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
35.0ms
(* (* x x) (+ 1 (cos x)))
25.0ms
(* (sin x) (sin x))
14.0ms
(+ 1 (cos x))

simplify4.3s

Counts
51 → 96
Calls

51 calls. Slowest were:

439.0ms
(/ (* (* (* (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)))))
378.0ms
(/ (* (* (* (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)))))
355.0ms
(- (+ (log (sin x)) (log (sin x))) (+ (log (* x x)) (log (+ 1 (cos x)))))

prune855.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0b

localize45.0ms

Local error

Found 4 expressions with local error:

15.0b
(/ (* (sin x) (sin x)) (/ (* (* x x) (+ (pow 1 3) (pow (cos x) 3))) (- (fma (cos x) (cos x) 1) (cos x))))
0.4b
(pow (cos x) 3)
0.3b
(+ (pow 1 3) (pow (cos x) 3))
0.2b
(* (sin x) (sin x))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 116
Calls

4 calls. Slowest were:

21.0ms
(/ (* (sin x) (sin x)) (/ (* (* x x) (+ (pow 1 3) (pow (cos x) 3))) (- (fma (cos x) (cos x) 1) (cos x))))
4.0ms
(+ (pow 1 3) (pow (cos x) 3))
4.0ms
(* (sin x) (sin x))

series158.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

106.0ms
(/ (* (sin x) (sin x)) (/ (* (* x x) (+ (pow 1 3) (pow (cos x) 3))) (- (fma (cos x) (cos x) 1) (cos x))))
20.0ms
(+ (pow 1 3) (pow (cos x) 3))
17.0ms
(pow (cos x) 3)
15.0ms
(* (sin x) (sin x))

simplify10.7s

Counts
91 → 128
Calls

91 calls. Slowest were:

874.0ms
(/ (* (* (* (sin x) (sin x)) (* (sin x) (sin x))) (* (sin x) (sin x))) (/ (* (* (* (* x x) (* x x)) (* x x)) (* (* (+ (pow 1 3) (pow (cos x) 3)) (+ (pow 1 3) (pow (cos x) 3))) (+ (pow 1 3) (pow (cos x) 3)))) (* (* (- (fma (cos x) (cos x) 1) (cos x)) (- (fma (cos x) (cos x) 1) (cos x))) (- (fma (cos x) (cos x) 1) (cos x)))))
641.0ms
(+ (* (fma (cos x) (cos x) 1) (fma (cos x) (cos x) 1)) (+ (* (cos x) (cos x)) (* (fma (cos x) (cos x) 1) (cos x))))
488.0ms
(/ (sin x) (+ (* (fma (cos x) (cos x) 1) (fma (cos x) (cos x) 1)) (+ (* (cos x) (cos x)) (* (fma (cos x) (cos x) 1) (cos x)))))

prune1.6s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize36.0ms

Local error

Found 4 expressions with local error:

15.0b
(/ (* (sin x) (sin x)) (* (* x x) (/ (+ (pow 1 3) (pow (cos x) 3)) (+ (* 1 1) (- (* (cos x) (cos x)) (* 1 (cos x)))))))
2.7b
(- (* (cos x) (cos x)) (* 1 (cos x)))
0.4b
(pow (cos x) 3)
0.3b
(+ (pow 1 3) (pow (cos x) 3))

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

21.0ms
(/ (* (sin x) (sin x)) (* (* x x) (/ (+ (pow 1 3) (pow (cos x) 3)) (+ (* 1 1) (- (* (cos x) (cos x)) (* 1 (cos x)))))))
6.0ms
(- (* (cos x) (cos x)) (* 1 (cos x)))
5.0ms
(+ (pow 1 3) (pow (cos x) 3))

series177.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

95.0ms
(/ (* (sin x) (sin x)) (* (* x x) (/ (+ (pow 1 3) (pow (cos x) 3)) (+ (* 1 1) (- (* (cos x) (cos x)) (* 1 (cos x)))))))
43.0ms
(- (* (cos x) (cos x)) (* 1 (cos x)))
22.0ms
(+ (pow 1 3) (pow (cos x) 3))
16.0ms
(pow (cos x) 3)

simplify4.6s

Counts
54 → 102
Calls

54 calls. Slowest were:

378.0ms
(/ (* (* (* (sin x) (sin x)) (sin x)) (* (* (sin x) (sin x)) (sin x))) (* (* (* (* x x) (/ (+ (pow 1 3) (pow (cos x) 3)) (+ (* 1 1) (- (* (cos x) (cos x)) (* 1 (cos x)))))) (* (* x x) (/ (+ (pow 1 3) (pow (cos x) 3)) (+ (* 1 1) (- (* (cos x) (cos x)) (* 1 (cos x))))))) (* (* x x) (/ (+ (pow 1 3) (pow (cos x) 3)) (+ (* 1 1) (- (* (cos x) (cos x)) (* 1 (cos x))))))))
368.0ms
(/ (* (pow (sin x) 2) (- (+ (pow (cos x) 2) 1) (cos x))) (* (pow x 2) (+ (pow (cos x) 3) 1)))
351.0ms
(- (log (* (sin x) (sin x))) (+ (log (* x x)) (log (/ (+ (pow 1 3) (pow (cos x) 3)) (+ (* 1 1) (- (* (cos x) (cos x)) (* 1 (cos x))))))))

prune1.4s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

regimes76.0ms

Accuracy

98.6% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 15.4b

bsearch82.0ms

end0.0ms

sample5.3s

Algorithm
intervals