Average Error: 1.0 → 0.1
Time: 11.8s
Precision: 64
Internal Precision: 128
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) \cdot \sin \left(\frac{\pi}{3}\right) + \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) \cdot \frac{1}{2}\right)\]

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019004 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))

Details

Time bar (total: 11.2s)Debug log

sample176.0ms

Algorithm
intervals

simplify12.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.0b

localize55.0ms

Local error

Found 4 expressions with local error:

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

rewrite34.0ms

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))
6.0ms
(/ (acos (/ (- g) h)) 3)

series172.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

120.0ms
(cos (+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2)))
39.0ms
(+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2))
12.0ms
(/ (acos (/ (- g) h)) 3)
0.0ms
(/ PI 3/2)

simplify329.0ms

Counts
23 → 60
Calls

23 calls. Slowest were:

54.0ms
(* (cos (/ (/ PI 2) 3)) (cos (- (/ (asin (/ (- g) h)) 3) (/ PI 3/2))))
37.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
35.0ms
(+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI))

prune646.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize50.0ms

Local error

Found 4 expressions with local error:

56.7b
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
1.0b
(sin (/ PI 3))
1.0b
(/ PI 3)
1.0b
(/ PI 3/2)

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 38
Calls

4 calls. Slowest were:

14.0ms
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
3.0ms
(sin (/ PI 3))
1.0ms
(/ PI 3)

series116.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

116.0ms
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
0.0ms
(sin (/ PI 3))
0.0ms
(/ PI 3)
0.0ms
(/ PI 3/2)

simplify666.0ms

Counts
10 → 44
Calls

10 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)
201.0ms
(* (+ (* (cos (- (/ (acos (/ g h)) 3) (/ PI 3/2))) 1/2) (* (sin (/ PI 3)) (sin (- (/ (acos (/ g h)) 3) (/ PI 3/2))))) 2)
171.0ms
(* (+ (* (cos (- (/ (acos (/ g h)) 3) (/ PI 3/2))) 1/2) (* (sin (/ PI 3)) (sin (- (/ (acos (/ g h)) 3) (/ PI 3/2))))) 2)

prune575.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 4 expressions with local error:

1.0b
(cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))
0.9b
(+ (* (acos (/ (- g) h)) 3/2) (* 3 PI))
0.0b
(* (acos (/ (- g) h)) 3/2)
0.0b
(/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 44
Calls

4 calls. Slowest were:

24.0ms
(/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)
10.0ms
(cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))
8.0ms
(+ (* (acos (/ (- g) h)) 3/2) (* 3 PI))

series194.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

104.0ms
(cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))
40.0ms
(/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)
38.0ms
(+ (* (acos (/ (- g) h)) 3/2) (* 3 PI))
12.0ms
(* (acos (/ (- g) h)) 3/2)

simplify830.0ms

Counts
19 → 56
Calls

19 calls. Slowest were:

271.0ms
(- (* (* (acos (/ (- g) h)) 3/2) (* (acos (/ (- g) h)) 3/2)) (* (* 3 PI) (* 3 PI)))
126.0ms
(+ (pow (* (acos (/ (- g) h)) 3/2) 3) (pow (* 3 PI) 3))
63.0ms
(* 2/9 (+ (* 3/2 (acos (* -1 (/ g h)))) (* 3 PI)))

prune547.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 4 expressions with local error:

1.0b
(cbrt (* (* (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)) (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))) (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))))
1.0b
(* (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)) (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)))
1.0b
(cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))
1.0b
(cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 47
Calls

4 calls. Slowest were:

7.0ms
(cbrt (* (* (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)) (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))) (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))))
6.0ms
(* (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)) (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)))
6.0ms
(cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))

series455.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

136.0ms
(cbrt (* (* (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)) (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))) (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))))
127.0ms
(* (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)) (cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2)))
108.0ms
(cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))
83.0ms
(cos (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))

simplify872.0ms

Counts
28 → 59
Calls

28 calls. Slowest were:

81.0ms
(* 1/2 (cos (* 2 (/ (+ (* (acos (/ (- g) h)) 3/2) (* 3 PI)) 9/2))))
68.0ms
(pow (cos (* 2/9 (+ (* 3/2 (acos (* -1 (/ g h)))) (* 3 PI)))) 2)
63.0ms
(cos (* 2/9 (+ (* 3/2 (acos (* -1 (/ g h)))) (* 3 PI))))

prune571.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

regimes337.0ms

Accuracy

0% (0.0b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch2.0ms

end0.0ms

sample4.4s

Algorithm
intervals