Average Error: 1.0 → 1.0
Time: 11.0s
Precision: 64
Internal Precision: 128
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right))_*\right) \cdot 2\]

Error

Bits error versus g

Bits error versus h

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

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

  3. Final simplification1.0

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

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))

Details

Time bar (total: 10.4s)Debug log

sample115.0ms

Algorithm
intervals

simplify20.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

20.0ms
(* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3))))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.0b

localize40.0ms

Local error

Found 3 expressions with local error:

1.0b
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
0.9b
(/ (acos (/ (- g) h)) 3)
0.0b
(fma 2/3 PI (/ (acos (/ (- g) h)) 3))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 37
Calls

3 calls. Slowest were:

4.0ms
(/ (acos (/ (- g) h)) 3)
4.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
0.0ms
(fma 2/3 PI (/ (acos (/ (- g) h)) 3))

series142.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

92.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
37.0ms
(fma 2/3 PI (/ (acos (/ (- g) h)) 3))
12.0ms
(/ (acos (/ (- g) h)) 3)

simplify275.0ms

Counts
14 → 46
Calls

14 calls. Slowest were:

41.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
38.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
37.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))

prune545.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 1.0b

localize10.0ms

Local error

Found 4 expressions with local error:

1.0b
(log1p (expm1 (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))
1.0b
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
0.9b
(expm1 (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))
0.9b
(/ (acos (/ (- g) h)) 3)

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 49
Calls

4 calls. Slowest were:

3.0ms
(/ (acos (/ (- g) h)) 3)
2.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
0.0ms
(expm1 (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))

series363.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

139.0ms
(log1p (expm1 (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))
116.0ms
(expm1 (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))
96.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
12.0ms
(/ (acos (/ (- g) h)) 3)

simplify387.0ms

Counts
19 → 61
Calls

19 calls. Slowest were:

52.0ms
(- (exp (cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))) 1)
39.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
38.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))

prune850.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 1.0b

localize22.0ms

Local error

Found 4 expressions with local error:

1.0b
(* (cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3))) (cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))
1.0b
(cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
1.0b
(cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
1.0b
(cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3))) (cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))
1.0ms
(cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
1.0ms
(cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))

series392.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

108.0ms
(cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
102.0ms
(* (cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3))) (cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))
98.0ms
(cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
83.0ms
(cbrt (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))

simplify1.7s

Counts
41 → 81
Calls

41 calls. Slowest were:

325.0ms
(pow (pow (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)) 2) 1/3)
292.0ms
(pow (pow (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)) 2) 1/3)
269.0ms
(pow (pow (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)) 2) 1/3)

prune1.1s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 1.0b

regimes187.0ms

Accuracy

0% (0.0b remaining)

Error of 1.0b against oracle of 1.0b and baseline of 1.0b

bsearch2.0ms

end0.0ms

sample4.2s

Algorithm
intervals