Average Error: 1.0 → 0.1
Time: 12.0s
Precision: 64
Internal Precision: 128
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\sqrt[3]{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right))_*\right)} \cdot \sqrt[3]{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right))_*\right) \cdot \cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right))_*\right)}\right)\]

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. Using strategy rm

  4. Applied add-cube-cbrt1.0

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

  6. Applied cbrt-unprod0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019002 +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: 11.4s)Debug log

sample161.0ms

Algorithm
intervals

simplify45.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

45.0ms
(* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3))))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.0b

localize30.0ms

Local error

Found 3 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 37
Calls

3 calls. Slowest were:

6.0ms
(/ (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))

series139.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

108.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
24.0ms
(fma 2/3 PI (/ (acos (/ (- g) h)) 3))
7.0ms
(/ (acos (/ (- g) h)) 3)

simplify223.0ms

Counts
14 → 46
Calls

14 calls. Slowest were:

37.0ms
(+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI))
36.0ms
(+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI))
22.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))

prune472.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 1.0b

localize9.0ms

Local error

Found 4 expressions with local error:

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

series393.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

153.0ms
(expm1 (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))
150.0ms
(log1p (expm1 (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))
78.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
12.0ms
(/ (acos (/ (- g) h)) 3)

simplify417.0ms

Counts
19 → 61
Calls

19 calls. Slowest were:

49.0ms
(- (exp (cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))) 1)
41.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
39.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))

prune825.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 1.0b

localize23.0ms

Local error

Found 4 expressions with local error:

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

10.0ms
(cbrt (* (* (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))) (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))) (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))
7.0ms
(* (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))) (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))
2.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))

series389.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

117.0ms
(cbrt (* (* (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))) (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))) (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))
95.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
93.0ms
(* (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))) (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))
85.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))

simplify790.0ms

Counts
30 → 69
Calls

30 calls. Slowest were:

119.0ms
(cbrt (* (+ (cos (+ (fma 2/3 PI (/ (acos (/ (- g) h)) 3)) (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))) (cos (- (fma 2/3 PI (/ (acos (/ (- g) h)) 3)) (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))) (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))
51.0ms
(* (* (* (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))) (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))) (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))) (* (* (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))) (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))) (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))
50.0ms
(* 1/2 (cos (* 2 (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))

prune1.1s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 1.0b

localize15.0ms

Local error

Found 4 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

11.0ms
(* (cbrt (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))) (cbrt (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))
3.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
3.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))

series508.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

209.0ms
(* (cbrt (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))) (cbrt (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))
118.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
96.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
84.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))

simplify592.0ms

Counts
35 → 72
Calls

35 calls. Slowest were:

37.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
35.0ms
(pow (pow (cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI))) 2) 1/3)
32.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))

prune938.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

regimes248.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample4.0s

Algorithm
intervals