Average Error: 1.0 → 0.0
Time: 12.7s
Precision: 64
Internal Precision: 128
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\frac{1}{2} \cdot \cos \left(\frac{\pi}{6} - \left(\frac{\pi}{\frac{3}{2}} + \frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right)\right) + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \sqrt[3]{\frac{\pi}{\frac{3}{2}}} \cdot \left(\sqrt[3]{\frac{\pi}{\frac{3}{2}}} \cdot \sqrt[3]{\frac{\pi}{\frac{3}{2}}}\right)\right)\right)\]

Error

Bits error versus g

Bits error versus h

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]

  2. Simplified1.0

    \[\leadsto \color{blue}{\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3} + \frac{\pi}{\frac{3}{2}}\right) \cdot 2}\]
  3. Using strategy rm

  4. Applied distribute-frac-neg1.0

    \[\leadsto \cos \left(\frac{\cos^{-1} \color{blue}{\left(-\frac{g}{h}\right)}}{3} + \frac{\pi}{\frac{3}{2}}\right) \cdot 2\]

  5. Applied acos-neg1.0

    \[\leadsto \cos \left(\frac{\color{blue}{\pi - \cos^{-1} \left(\frac{g}{h}\right)}}{3} + \frac{\pi}{\frac{3}{2}}\right) \cdot 2\]

  6. Applied div-sub1.0

    \[\leadsto \cos \left(\color{blue}{\left(\frac{\pi}{3} - \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)} + \frac{\pi}{\frac{3}{2}}\right) \cdot 2\]

  7. Applied associate-+l-1.0

    \[\leadsto \cos \color{blue}{\left(\frac{\pi}{3} - \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right)} \cdot 2\]

  8. Applied cos-diff0.1

    \[\leadsto \color{blue}{\left(\cos \left(\frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right)} \cdot 2\]

  9. Simplified0.1

    \[\leadsto \left(\color{blue}{\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) \cdot \frac{1}{2}} + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right) \cdot 2\]
  10. Using strategy rm

  11. Applied acos-asin0.1

    \[\leadsto \left(\cos \left(\frac{\color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{g}{h}\right)}}{3} - \frac{\pi}{\frac{3}{2}}\right) \cdot \frac{1}{2} + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right) \cdot 2\]

  12. Applied div-sub0.1

    \[\leadsto \left(\cos \left(\color{blue}{\left(\frac{\frac{\pi}{2}}{3} - \frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right)} - \frac{\pi}{\frac{3}{2}}\right) \cdot \frac{1}{2} + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right) \cdot 2\]

  13. Applied associate--l-0.0

    \[\leadsto \left(\cos \color{blue}{\left(\frac{\frac{\pi}{2}}{3} - \left(\frac{\sin^{-1} \left(\frac{g}{h}\right)}{3} + \frac{\pi}{\frac{3}{2}}\right)\right)} \cdot \frac{1}{2} + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right) \cdot 2\]

  14. Simplified0.0

    \[\leadsto \left(\cos \left(\color{blue}{\frac{\pi}{6}} - \left(\frac{\sin^{-1} \left(\frac{g}{h}\right)}{3} + \frac{\pi}{\frac{3}{2}}\right)\right) \cdot \frac{1}{2} + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right) \cdot 2\]
  15. Using strategy rm

  16. Applied add-cube-cbrt0.0

    \[\leadsto \left(\cos \left(\frac{\pi}{6} - \left(\frac{\sin^{-1} \left(\frac{g}{h}\right)}{3} + \frac{\pi}{\frac{3}{2}}\right)\right) \cdot \frac{1}{2} + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \color{blue}{\left(\sqrt[3]{\frac{\pi}{\frac{3}{2}}} \cdot \sqrt[3]{\frac{\pi}{\frac{3}{2}}}\right) \cdot \sqrt[3]{\frac{\pi}{\frac{3}{2}}}}\right)\right) \cdot 2\]

  17. Final simplification0.0

    \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot \cos \left(\frac{\pi}{6} - \left(\frac{\pi}{\frac{3}{2}} + \frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right)\right) + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \sqrt[3]{\frac{\pi}{\frac{3}{2}}} \cdot \left(\sqrt[3]{\frac{\pi}{\frac{3}{2}}} \cdot \sqrt[3]{\frac{\pi}{\frac{3}{2}}}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019002 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))

Details

Time bar (total: 12.0s)Debug log

sample183.0ms

Algorithm
intervals

simplify12.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

12.0ms
(* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3))))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.0b

localize77.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ PI 3/2)
1.0b
(+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2))
1.0b
(cos (+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2)))
1.0b
(/ (acos (/ (- g) h)) 3)

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

16.0ms
(cos (+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2)))
10.0ms
(+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2))
6.0ms
(/ (acos (/ (- g) h)) 3)

series162.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

116.0ms
(cos (+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2)))
38.0ms
(+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2))
8.0ms
(/ (acos (/ (- g) h)) 3)
0.0ms
(/ PI 3/2)

simplify362.0ms

Counts
23 → 60
Calls

23 calls. Slowest were:

38.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
38.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
36.0ms
(+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI))

prune692.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize66.0ms

Local error

Found 4 expressions with local error:

56.8b
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
1.0b
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
1.0b
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
1.0b
(sin (/ PI 3))

rewrite39.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

15.0ms
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
11.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
10.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))

series186.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

101.0ms
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
43.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
42.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
0.0ms
(sin (/ PI 3))

simplify478.0ms

Counts
28 → 60
Calls

28 calls. Slowest were:

223.0ms
(* (+ (* (cos (- (/ (acos (/ g h)) 3) (/ PI 3/2))) 1/2) (* (sin (/ PI 3)) (sin (- (/ (acos (/ g h)) 3) (/ PI 3/2))))) 2)
35.0ms
(/ (exp (/ (acos (/ g h)) 3)) (exp (/ PI 3/2)))
33.0ms
(/ (exp (/ (acos (/ g h)) 3)) (exp (/ PI 3/2)))

prune918.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

localize57.0ms

Local error

Found 4 expressions with local error:

56.8b
(cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2))))
1.0b
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
1.0b
(sin (/ PI 3))
1.0b
(/ PI 3/2)

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
4 → 45
Calls

4 calls. Slowest were:

33.0ms
(cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2))))
5.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
1.0ms
(sin (/ PI 3))

series129.0ms

Counts
4 → 8
Calls

4 calls. Slowest were:

100.0ms
(cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2))))
29.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
0.0ms
(sin (/ PI 3))
0.0ms
(/ PI 3/2)

simplify675.0ms

Counts
20 → 53
Calls

20 calls. Slowest were:

219.0ms
(* (+ (* (cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2)))) 1/2) (* (sin (/ PI 3)) (sin (- (/ (acos (/ g h)) 3) (/ PI 3/2))))) 2)
216.0ms
(* (+ (* (cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2)))) 1/2) (* (sin (/ PI 3)) (sin (- (/ (acos (/ g h)) 3) (/ PI 3/2))))) 2)
32.0ms
(* (cos (/ PI 6)) (cos (- (+ (/ (asin (/ g h)) 3) (/ PI 3/2)))))

prune1.0s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize60.0ms

Local error

Found 4 expressions with local error:

56.8b
(cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2))))
1.0b
(- (/ (acos (/ g h)) 3) (* (* (cbrt (/ PI 3/2)) (cbrt (/ PI 3/2))) (cbrt (/ PI 3/2))))
1.0b
(sin (/ PI 3))
1.0b
(/ PI 3/2)

rewrite71.0ms

Algorithm
rewrite-expression-head
Counts
4 → 49
Calls

4 calls. Slowest were:

44.0ms
(cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2))))
22.0ms
(- (/ (acos (/ g h)) 3) (* (* (cbrt (/ PI 3/2)) (cbrt (/ PI 3/2))) (cbrt (/ PI 3/2))))
3.0ms
(sin (/ PI 3))

series215.0ms

Counts
4 → 8
Calls

4 calls. Slowest were:

112.0ms
(- (/ (acos (/ g h)) 3) (* (* (cbrt (/ PI 3/2)) (cbrt (/ PI 3/2))) (cbrt (/ PI 3/2))))
104.0ms
(cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2))))
0.0ms
(sin (/ PI 3))
0.0ms
(/ PI 3/2)

simplify1.1s

Counts
27 → 57
Calls

27 calls. Slowest were:

149.0ms
(- (* (acos (/ g h)) (* (cbrt 3/2) (cbrt 3/2))) (* 3 (* (* (cbrt PI) (cbrt (/ PI 3/2))) (cbrt PI))))
131.0ms
(- (* (acos (/ g h)) (* (cbrt 3/2) (cbrt 3/2))) (* 3 (* (* (cbrt (/ PI 3/2)) (cbrt PI)) (cbrt PI))))
115.0ms
(- (* (acos (/ g h)) (cbrt 3/2)) (* 3 (* (* (cbrt PI) (cbrt (/ PI 3/2))) (cbrt (/ PI 3/2)))))

prune1.2s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

regimes406.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample3.9s

Algorithm
intervals