Average Error: 1.0 → 0.0
Time: 10.1s
Precision: 64
Internal Precision: 128
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\left(\cos \left(\frac{\pi}{\frac{3}{2}}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\frac{\pi}{\frac{3}{2}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right) \cdot \frac{1}{2} + \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) \cdot \sin \left(\sqrt{\frac{\pi}{3}} \cdot \sqrt{\frac{\pi}{3}}\right)\right) \cdot 2\]

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 add-sqr-sqrt1.0

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

  13. Applied cos-diff0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2019005 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))

Details

Time bar (total: 9.4s)Debug log

sample163.0ms

Algorithm
intervals

simplify41.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

41.0ms
(* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3))))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.0b

localize30.0ms

Local error

Found 4 expressions with local error:

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

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

8.0ms
(cos (+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2)))
5.0ms
(+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2))
3.0ms
(/ (acos (/ (- g) h)) 3)

series121.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

88.0ms
(cos (+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2)))
25.0ms
(+ (/ (acos (/ (- g) h)) 3) (/ PI 3/2))
8.0ms
(/ (acos (/ (- g) h)) 3)
0.0ms
(/ PI 3/2)

simplify302.0ms

Counts
23 → 60
Calls

23 calls. Slowest were:

32.0ms
(+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI))
29.0ms
(+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI))
29.0ms
(* (cos (/ (/ PI 2) 3)) (cos (- (/ (asin (/ (- g) h)) 3) (/ PI 3/2))))

prune550.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize43.0ms

Local error

Found 4 expressions with local error:

56.4b
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
1.0b
(sin (/ PI 3))
1.0b
(/ PI 3)
1.0b
(/ PI 3/2)

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 38
Calls

4 calls. Slowest were:

8.0ms
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
1.0ms
(sin (/ PI 3))
0.0ms
(/ PI 3/2)

series82.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

82.0ms
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
0.0ms
(sin (/ PI 3))
0.0ms
(/ PI 3/2)
0.0ms
(/ PI 3)

simplify753.0ms

Counts
10 → 44
Calls

10 calls. Slowest were:

256.0ms
(* (+ (* (cos (- (/ (acos (/ g h)) 3) (/ PI 3/2))) 1/2) (* (sin (/ PI 3)) (sin (- (/ (acos (/ g h)) 3) (/ PI 3/2))))) 2)
224.0ms
(* (+ (* (cos (- (/ (acos (/ g h)) 3) (/ PI 3/2))) 1/2) (* (sin (/ PI 3)) (sin (- (/ (acos (/ g h)) 3) (/ PI 3/2))))) 2)
223.0ms
(* (+ (* (cos (- (/ (acos (/ g h)) 3) (/ PI 3/2))) 1/2) (* (sin (/ PI 3)) (sin (- (/ (acos (/ g h)) 3) (/ PI 3/2))))) 2)

prune685.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize31.0ms

Local error

Found 4 expressions with local error:

56.4b
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
1.0b
(/ PI 3/2)
1.0b
(/ PI 3/2)
1.0b
(sin (* (sqrt (/ PI 3)) (sqrt (/ PI 3))))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
4 → 37
Calls

4 calls. Slowest were:

16.0ms
(sin (* (sqrt (/ PI 3)) (sqrt (/ PI 3))))
15.0ms
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
0.0ms
(/ PI 3/2)

series105.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

104.0ms
(cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))
0.0ms
(/ PI 3/2)
0.0ms
(/ PI 3/2)
0.0ms
(sin (* (sqrt (/ PI 3)) (sqrt (/ PI 3))))

simplify59.0ms

Counts
6 → 43
Calls

6 calls. Slowest were:

15.0ms
(cos (- (* 1/3 (acos (/ g h))) (* 2/3 PI)))
14.0ms
(cos (- (* 1/3 (acos (/ g h))) (* 2/3 PI)))
14.0ms
(cos (- (* 1/3 (acos (/ g h))) (* 2/3 PI)))

prune749.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize62.0ms

Local error

Found 4 expressions with local error:

59.6b
(+ (* (cos (/ (acos (/ g h)) 3)) (cos (/ PI 3/2))) (* (sin (/ (acos (/ g h)) 3)) (sin (/ PI 3/2))))
1.6b
(cos (/ PI 3/2))
1.0b
(sin (/ PI 3/2))
1.0b
(/ PI 3/2)

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 36
Calls

4 calls. Slowest were:

17.0ms
(+ (* (cos (/ (acos (/ g h)) 3)) (cos (/ PI 3/2))) (* (sin (/ (acos (/ g h)) 3)) (sin (/ PI 3/2))))
3.0ms
(cos (/ PI 3/2))
2.0ms
(sin (/ PI 3/2))

series348.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

348.0ms
(+ (* (cos (/ (acos (/ g h)) 3)) (cos (/ PI 3/2))) (* (sin (/ (acos (/ g h)) 3)) (sin (/ PI 3/2))))
0.0ms
(cos (/ PI 3/2))
0.0ms
(/ PI 3/2)
0.0ms
(sin (/ PI 3/2))

simplify233.0ms

Counts
9 → 42
Calls

9 calls. Slowest were:

175.0ms
(+ (* (+ (cos (+ (/ (acos (/ g h)) 3) (/ PI 3/2))) (cos (- (/ (acos (/ g h)) 3) (/ PI 3/2)))) 2) (* 2 (- (cos (- (/ (acos (/ g h)) 3) (/ PI 3/2))) (cos (+ (/ (acos (/ g h)) 3) (/ PI 3/2))))))
14.0ms
(* (exp (* (cos (/ (acos (/ g h)) 3)) (cos (/ PI 3/2)))) (exp (* (sin (/ (acos (/ g h)) 3)) (sin (/ PI 3/2)))))
7.0ms
(* (cos (/ (acos (/ g h)) 3)) (cos (/ PI 3/2)))

prune866.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

regimes105.0ms

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