Average Error: 1.0 → 0.0
Time: 11.5s
Precision: 64
Internal Precision: 128
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\frac{1}{2} \cdot \cos \left(\frac{\pi}{6} - \left(\frac{\pi}{\frac{3}{2}} + \frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right)\right) + \sin \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right)\right)\]

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-sub0.0

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

  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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019016 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))

Details

Time bar (total: 10.9s)Debug log

sample204.0ms

Algorithm
intervals
Results
544×(pre true 80)
288×(body nan 80)
256×(body real 80)

simplify12.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
12.0ms
(* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3))))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.0b

localize32.0ms

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)))
0.9b
(/ (acos (/ (- g) h)) 3)

rewrite43.0ms

Algorithm
rewrite-expression-head
Rules
7×*-un-lft-identity
6×add-log-exp
6×div-sub
5×add-cube-cbrt
5×add-sqr-sqrt
4×associate-+l-
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×associate-/l*
3×distribute-frac-neg
3×acos-neg
3×acos-asin
2×div-inv
2×frac-2neg
2×clear-num
2×cos-diff
1×distribute-lft-out
1×flip-+
1×frac-add
1×flip3-+
1×sum-log
1×cos-sum
1×+-commutative
Counts
4 → 50
Calls
4 calls:
Slowest
26.0ms
(cos (+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2)))
10.0ms
(+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2))
6.0ms
(/ (acos (/ (- g) h)) 3)
0.0ms
(/ PI 3/2)

series169.0ms

Counts
4 → 10
Calls
4 calls:
Slowest
118.0ms
(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)

simplify315.0ms

Counts
23 → 60
Calls
23 calls:
Slowest
43.0ms
(* (cos (/ (/ PI 2) 3)) (cos (- (/ (asin (/ (- g) h)) 3) (/ PI 3/2))))
34.0ms
(+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI))
28.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
25.0ms
(* 1/3 (acos (* -1 (/ g h))))
23.0ms
(+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI))

prune692.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize72.0ms

Local error

Found 4 expressions with local error:

56.4b
(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))

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
8×add-log-exp
8×*-un-lft-identity
8×add-sqr-sqrt
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×sub-neg
2×difference-of-squares
2×distribute-lft-out--
2×flip--
2×frac-sub
2×div-sub
2×diff-log
2×flip3--
2×cos-sum
2×acos-asin
2×associate--l-
1×sin-PI/3
1×cos-diff
Counts
4 → 50
Calls
4 calls:
Slowest
16.0ms
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
11.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
9.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
1.0ms
(sin (/ PI 3))

series156.0ms

Counts
4 → 10
Calls
4 calls:
Slowest
95.0ms
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
34.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
27.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
0.0ms
(sin (/ PI 3))

simplify413.0ms

Counts
28 → 60
Calls
28 calls:
Slowest
168.0ms
(* (+ (* (cos (- (/ (acos (/ g h)) 3) (/ PI 3/2))) 1/2) (* (sin (/ PI 3)) (sin (- (/ (acos (/ g h)) 3) (/ PI 3/2))))) 2)
33.0ms
(/ (exp (/ (acos (/ g h)) 3)) (exp (/ PI 3/2)))
16.0ms
(/ (exp (/ (acos (/ g h)) 3)) (exp (/ PI 3/2)))
15.0ms
(cos (- (* 1/3 (acos (/ g h))) (* 2/3 PI)))
14.0ms
(- (* (acos (/ g h)) 3/2) (* 3 PI))

prune878.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize56.0ms

Local error

Found 4 expressions with local error:

56.4b
(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
Rules
6×add-log-exp
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×cos-sum
2×div-sub
2×sub-neg
1×difference-of-squares
1×associate--r-
1×distribute-lft-out--
1×flip--
1×div-inv
1×associate-+l-
1×frac-sub
1×diff-log
1×flip3--
1×frac-2neg
1×sin-PI/3
1×clear-num
1×asin-acos
1×cos-diff
1×acos-asin
1×associate--l-
Counts
4 → 45
Calls
4 calls:
Slowest
32.0ms
(cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2))))
5.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
1.0ms
(sin (/ PI 3))
0.0ms
(/ PI 3/2)

series149.0ms

Counts
4 → 8
Calls
4 calls:
Slowest
111.0ms
(cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2))))
38.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
0.0ms
(sin (/ PI 3))
0.0ms
(/ PI 3/2)

simplify699.0ms

Counts
20 → 53
Calls
20 calls:
Slowest
209.0ms
(* (+ (* (cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2)))) 1/2) (* (sin (/ PI 3)) (sin (- (/ (acos (/ g h)) 3) (/ PI 3/2))))) 2)
155.0ms
(* (+ (* (cos (- (/ PI 6) (+ (/ (asin (/ g h)) 3) (/ PI 3/2)))) 1/2) (* (sin (/ PI 3)) (sin (- (/ (acos (/ g h)) 3) (/ PI 3/2))))) 2)
55.0ms
(* (cos (/ PI 6)) (cos (- (+ (/ (asin (/ g h)) 3) (/ PI 3/2)))))
44.0ms
(* (cos (- (/ PI 6) (/ (/ PI 2) 3))) (cos (- (/ (acos (/ g h)) 3) (/ PI 3/2))))
41.0ms
(* (cos (/ PI 6)) (cos (+ (/ (asin (/ g h)) 3) (/ PI 3/2))))

prune871.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize22.0ms

Local error

Found 4 expressions with local error:

56.4b
(cos (cbrt (* (* (- (/ (acos (/ g h)) 3) (/ PI 3/2)) (- (/ (acos (/ g h)) 3) (/ PI 3/2))) (- (/ (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
(- (/ (acos (/ g h)) 3) (/ PI 3/2))

rewrite62.0ms

Algorithm
rewrite-expression-head
Rules
10×add-log-exp
10×*-un-lft-identity
10×add-sqr-sqrt
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×difference-of-squares
3×distribute-lft-out--
3×flip--
3×frac-sub
3×div-sub
3×diff-log
3×flip3--
3×sub-neg
3×acos-asin
3×associate--l-
Counts
4 → 55
Calls
4 calls:
Slowest
35.0ms
(cos (cbrt (* (* (- (/ (acos (/ g h)) 3) (/ PI 3/2)) (- (/ (acos (/ g h)) 3) (/ PI 3/2))) (- (/ (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))
6.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))

series212.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
94.0ms
(cos (cbrt (* (* (- (/ (acos (/ g h)) 3) (/ PI 3/2)) (- (/ (acos (/ g h)) 3) (/ PI 3/2))) (- (/ (acos (/ g h)) 3) (/ PI 3/2)))))
42.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
39.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))
36.0ms
(- (/ (acos (/ g h)) 3) (/ PI 3/2))

simplify343.0ms

Counts
33 → 67
Calls
33 calls:
Slowest
33.0ms
(/ (exp (/ (acos (/ g h)) 3)) (exp (/ PI 3/2)))
33.0ms
(/ (exp (/ (acos (/ g h)) 3)) (exp (/ PI 3/2)))
18.0ms
(/ (exp (/ (acos (/ g h)) 3)) (exp (/ PI 3/2)))
15.0ms
(- (* (acos (/ g h)) 3/2) (* 3 PI))
14.0ms
(cos (- (* 1/3 (acos (/ g h))) (* 2/3 PI)))

prune1.0s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes234.0ms

Accuracy

0% (0.0b remaining)

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

bsearch6.0ms

end0.0ms

sample4.2s

Algorithm
intervals
Results
15971×(pre true 80)
8001×(body real 80)
7970×(body nan 80)