Average Error: 31.4 → 0.4
Time: 42.5s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{\left|\frac{x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + \left(1 - \cos x\right))_*}}\right|} \cdot \frac{\sin x}{\left|\frac{x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + \left(1 - \cos x\right))_*}}\right|}}{\frac{1 + {\left(\cos x\right)}^{3}}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}\]

Error

Bits error versus x

Derivation

  1. Initial program 31.4

    \[\frac{1 - \cos x}{x \cdot x}\]

  2. Initial simplification31.4

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

  4. Applied flip--31.6

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

  5. Applied associate-/l/31.6

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

  6. Simplified15.2

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

  8. Applied flip3-+15.2

    \[\leadsto \frac{\sin x \cdot \sin x}{\left(x \cdot x\right) \cdot \color{blue}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)}}}\]

  9. Applied associate-*r/15.2

    \[\leadsto \frac{\sin x \cdot \sin x}{\color{blue}{\frac{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}{1 \cdot 1 + \left(\cos x \cdot \cos x - 1 \cdot \cos x\right)}}}\]

  10. Simplified15.2

    \[\leadsto \frac{\sin x \cdot \sin x}{\frac{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}{\color{blue}{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt15.4

    \[\leadsto \frac{\sin x \cdot \sin x}{\frac{\left(x \cdot x\right) \cdot \left({1}^{3} + {\left(\cos x\right)}^{3}\right)}{\color{blue}{\left(\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x} \cdot \sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}\right) \cdot \sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}}\]

  13. Applied times-frac15.4

    \[\leadsto \frac{\sin x \cdot \sin x}{\color{blue}{\frac{x \cdot x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x} \cdot \sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}} \cdot \frac{{1}^{3} + {\left(\cos x\right)}^{3}}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}}\]

  14. Applied associate-/r*15.4

    \[\leadsto \color{blue}{\frac{\frac{\sin x \cdot \sin x}{\frac{x \cdot x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x} \cdot \sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}}\]
  15. Using strategy rm

  16. Applied add-sqr-sqrt15.4

    \[\leadsto \frac{\frac{\sin x \cdot \sin x}{\color{blue}{\sqrt{\frac{x \cdot x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x} \cdot \sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}} \cdot \sqrt{\frac{x \cdot x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x} \cdot \sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}}}}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}\]

  17. Applied times-frac15.7

    \[\leadsto \frac{\color{blue}{\frac{\sin x}{\sqrt{\frac{x \cdot x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x} \cdot \sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}} \cdot \frac{\sin x}{\sqrt{\frac{x \cdot x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x} \cdot \sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}}}}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}\]

  18. Simplified15.6

    \[\leadsto \frac{\color{blue}{\frac{\sin x}{\left|\frac{x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + \left(1 - \cos x\right))_*}}\right|}} \cdot \frac{\sin x}{\sqrt{\frac{x \cdot x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x} \cdot \sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}}}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}\]

  19. Simplified0.4

    \[\leadsto \frac{\frac{\sin x}{\left|\frac{x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + \left(1 - \cos x\right))_*}}\right|} \cdot \color{blue}{\frac{\sin x}{\left|\frac{x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + \left(1 - \cos x\right))_*}}\right|}}}{\frac{{1}^{3} + {\left(\cos x\right)}^{3}}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}\]

  20. Final simplification0.4

    \[\leadsto \frac{\frac{\sin x}{\left|\frac{x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + \left(1 - \cos x\right))_*}}\right|} \cdot \frac{\sin x}{\left|\frac{x}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + \left(1 - \cos x\right))_*}}\right|}}{\frac{1 + {\left(\cos x\right)}^{3}}{\sqrt[3]{(\left(\cos x\right) \cdot \left(\cos x\right) + 1)_* - \cos x}}}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))

Details

Time bar (total: 34.3s)Debug log

start213.0ms

Algorithm
intervals

setup43.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 33.4b

localize22.0ms

Local error

Found 2 expressions with local error:

18.5b
(/ (- 1 (cos x)) (* x x))
2.4b
(- 1 (cos x))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
2 → 37
Calls

2 calls. Slowest were:

8.0ms
(/ (- 1 (cos x)) (* x x))
2.0ms
(- 1 (cos x))

series89.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

68.0ms
(/ (- 1 (cos x)) (* x x))
20.0ms
(- 1 (cos x))

simplify761.0ms

Counts
19 → 43
Calls

19 calls. Slowest were:

414.0ms
(/ (* (* (- 1 (cos x)) (- 1 (cos x))) (- 1 (cos x))) (* (* (* x x) (* x x)) (* x x)))
100.0ms
(- (+ (* 1/2 (pow x 2)) (* 1/720 (pow x 6))) (* 1/24 (pow x 4)))
75.0ms
(- (+ (* 1/720 (pow x 4)) 1/2) (* 1/24 (pow x 2)))

prune435.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

18.2b
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
0.3b
(+ 1 (cos x))
0.3b
(* (sin x) (sin x))
0.1b
(* (* x x) (+ 1 (cos x)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

8.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
4.0ms
(* (sin x) (sin x))
4.0ms
(* (* x x) (+ 1 (cos x)))

series122.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

60.0ms
(/ (* (sin x) (sin x)) (* (* x x) (+ 1 (cos x))))
33.0ms
(* (* x x) (+ 1 (cos x)))
20.0ms
(* (sin x) (sin x))
8.0ms
(+ 1 (cos x))

simplify5.1s

Counts
51 → 96
Calls

51 calls. Slowest were:

1.2s
(* (* (* (* x x) (* x x)) (* x x)) (* (* (+ 1 (cos x)) (+ 1 (cos x))) (+ 1 (cos x))))
400.0ms
(- (+ (log (sin x)) (log (sin x))) (+ (log (* x x)) (log (+ 1 (cos x)))))
342.0ms
(- (log (* (sin x) (sin x))) (log (* (* x x) (+ 1 (cos x)))))

prune1.1s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

18.2b
(/ (* (sin x) (sin x)) (/ (* (* x x) (+ (pow 1 3) (pow (cos x) 3))) (- (fma (cos x) (cos x) 1) (cos x))))
0.3b
(pow (cos x) 3)
0.3b
(* (sin x) (sin x))
0.2b
(+ (pow 1 3) (pow (cos x) 3))

rewrite55.0ms

Algorithm
rewrite-expression-head
Counts
4 → 116
Calls

4 calls. Slowest were:

28.0ms
(/ (* (sin x) (sin x)) (/ (* (* x x) (+ (pow 1 3) (pow (cos x) 3))) (- (fma (cos x) (cos x) 1) (cos x))))
9.0ms
(+ (pow 1 3) (pow (cos x) 3))
7.0ms
(* (sin x) (sin x))

series212.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

135.0ms
(/ (* (sin x) (sin x)) (/ (* (* x x) (+ (pow 1 3) (pow (cos x) 3))) (- (fma (cos x) (cos x) 1) (cos x))))
30.0ms
(pow (cos x) 3)
24.0ms
(+ (pow 1 3) (pow (cos x) 3))
23.0ms
(* (sin x) (sin x))

simplify11.6s

Counts
91 → 128
Calls

91 calls. Slowest were:

722.0ms
(- (+ (log (sin x)) (log (sin x))) (log (/ (* (* x x) (+ (pow 1 3) (pow (cos x) 3))) (- (fma (cos x) (cos x) 1) (cos x)))))
556.0ms
(/ (* (* (* (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)))))
514.0ms
(- (log (* (sin x) (sin x))) (- (+ (log (* x x)) (log (+ (pow 1 3) (pow (cos x) 3)))) (log (- (fma (cos x) (cos x) 1) (cos x)))))

prune1.9s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

18.2b
(/ (* (sin x) (sin x)) (/ (* x x) (* (cbrt (- (fma (cos x) (cos x) 1) (cos x))) (cbrt (- (fma (cos x) (cos x) 1) (cos x))))))
0.3b
(pow (cos x) 3)
0.3b
(* (sin x) (sin x))
0.3b
(cbrt (- (fma (cos x) (cos x) 1) (cos x)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 116
Calls

4 calls. Slowest were:

17.0ms
(/ (* (sin x) (sin x)) (/ (* x x) (* (cbrt (- (fma (cos x) (cos x) 1) (cos x))) (cbrt (- (fma (cos x) (cos x) 1) (cos x))))))
6.0ms
(* (sin x) (sin x))
4.0ms
(cbrt (- (fma (cos x) (cos x) 1) (cos x)))

series548.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

277.0ms
(/ (* (sin x) (sin x)) (/ (* x x) (* (cbrt (- (fma (cos x) (cos x) 1) (cos x))) (cbrt (- (fma (cos x) (cos x) 1) (cos x))))))
218.0ms
(cbrt (- (fma (cos x) (cos x) 1) (cos x)))
27.0ms
(pow (cos x) 3)
26.0ms
(* (sin x) (sin x))

simplify9.6s

Counts
96 → 128
Calls

96 calls. Slowest were:

492.0ms
(/ (sin x) (/ (* x x) (* (cbrt (- (* (fma (cos x) (cos x) 1) (fma (cos x) (cos x) 1)) (* (cos x) (cos x)))) (cbrt (- (* (fma (cos x) (cos x) 1) (fma (cos x) (cos x) 1)) (* (cos x) (cos x)))))))
478.0ms
(- (log (* (sin x) (sin x))) (log (/ (* x x) (* (cbrt (- (fma (cos x) (cos x) 1) (cos x))) (cbrt (- (fma (cos x) (cos x) 1) (cos x)))))))
407.0ms
(cbrt (- (* (fma (cos x) (cos x) 1) (fma (cos x) (cos x) 1)) (* (cos x) (cos x))))

prune2.3s

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0b

regimes91.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch3.0ms