Average Error: 1.0 → 0.0
Time: 12.3s
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-cbrt-cube1.5

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

  6. Applied cbrt-prod0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019010 +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: 11.3s)Debug log

sample209.0ms

Algorithm
intervals

simplify32.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
32.0ms
(* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3))))

prune26.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.0b

localize46.0ms

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))

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×add-log-exp
3×associate-/l*
3×log1p-expm1-u
3×add-exp-log
3×add-cbrt-cube
3×pow1
3×expm1-log1p-u
2×fma-udef
2×div-sub
1×div-inv
1×distribute-frac-neg
1×frac-2neg
1×clear-num
1×cos-sum
1×acos-neg
1×acos-asin
Counts
3 → 37
Calls
3 calls:
Slowest
7.0ms
(/ (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))

series150.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
100.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
38.0ms
(fma 2/3 PI (/ (acos (/ (- g) h)) 3))
12.0ms
(/ (acos (/ (- g) h)) 3)

simplify258.0ms

Counts
14 → 46
Calls
14 calls:
Slowest
40.0ms
(+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI))
38.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
28.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
27.0ms
(+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI))
25.0ms
(* 1/3 (acos (* -1 (/ g h))))

prune565.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.9b

localize39.0ms

Local error

Found 4 expressions with local error:

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

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
20×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×pow1
7×associate-*l*
7×associate-*r*
7×pow1/3
7×cbrt-div
6×add-exp-log
5×div-inv
4×add-log-exp
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-plus
1×frac-times
1×pow2
Counts
4 → 80
Calls
4 calls:
Slowest
11.0ms
(* (cbrt (/ (acos (/ (- g) h)) 3)) (cbrt (/ (acos (/ (- g) h)) 3)))
1.0ms
(cbrt (/ (acos (/ (- g) h)) 3))
1.0ms
(cbrt (/ (acos (/ (- g) h)) 3))
1.0ms
(cbrt (/ (acos (/ (- g) h)) 3))

series412.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
149.0ms
(* (cbrt (/ (acos (/ (- g) h)) 3)) (cbrt (/ (acos (/ (- g) h)) 3)))
101.0ms
(cbrt (/ (acos (/ (- g) h)) 3))
86.0ms
(cbrt (/ (acos (/ (- g) h)) 3))
76.0ms
(cbrt (/ (acos (/ (- g) h)) 3))

simplify478.0ms

Counts
53 → 92
Calls
53 calls:
Slowest
36.0ms
(* (cbrt 1/3) (pow (acos (* -1 (/ g h))) 1/3))
31.0ms
(* (pow (cbrt 1/3) 2) (pow (pow (acos (* -1 (/ g h))) 2) 1/3))
30.0ms
(* (pow (cbrt 1/3) 2) (pow (pow (acos (* -1 (/ g h))) 2) 1/3))
28.0ms
(* (pow (cbrt 1/3) 2) (pow (pow (acos (* -1 (/ g h))) 2) 1/3))
25.0ms
(* (/ (acos (/ (- g) h)) 3) (/ (acos (/ (- g) h)) 3))

prune1.3s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.9b

localize22.0ms

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
5×log1p-expm1-u
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×associate-/l*
2×div-sub
1×expm1-udef
1×div-inv
1×fma-udef
1×distribute-frac-neg
1×log1p-expm1
1×frac-2neg
1×clear-num
1×expm1-log1p
1×log1p-udef
1×cos-sum
1×acos-neg
1×acos-asin
Counts
4 → 49
Calls
4 calls:
Slowest
7.0ms
(/ (acos (/ (- g) h)) 3)
4.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
1.0ms
(expm1 (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))
0.0ms
(log1p (expm1 (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))

series345.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
133.0ms
(log1p (expm1 (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))))
113.0ms
(expm1 (cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3))))
88.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
11.0ms
(/ (acos (/ (- g) h)) 3)

simplify328.0ms

Counts
19 → 61
Calls
19 calls:
Slowest
38.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
29.0ms
(- (exp (cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))) 1)
29.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
29.0ms
(- (exp (cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))) 1)
28.0ms
(- (exp (cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))) 1)

prune819.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.9b

localize24.0ms

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

rewrite34.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×log1p-expm1-u
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×expm1-log1p-u
4×add-sqr-sqrt
3×fma-udef
3×cos-sum
1×rem-cbrt-cube
1×cos-mult
1×associate-*l/
1×pow3
1×pow1/3
1×cbrt-div
1×cbrt-prod
Counts
4 → 43
Calls
4 calls:
Slowest
21.0ms
(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)))))
4.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
4.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
4.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))

series379.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
116.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
98.0ms
(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)))))
86.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))
79.0ms
(cos (fma 2/3 PI (/ (acos (/ (- g) h)) 3)))

simplify523.0ms

Counts
18 → 55
Calls
18 calls:
Slowest
86.0ms
(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)))))
40.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
39.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
39.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))
38.0ms
(cos (+ (* 1/3 (acos (* -1 (/ g h)))) (* 2/3 PI)))

prune691.0ms

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0b

regimes350.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample4.2s

Algorithm
intervals