Average Error: 10.0 → 10.0
Time: 57.9s
Precision: 64
Internal Precision: 128
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 10.0

    \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]

  2. Final simplification10.0

    \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]

Reproduce

herbie shell --seed 2019008 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))

Details

Time bar (total: 56.1s)Debug log

sample179.0ms

Algorithm
intervals

simplify211.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

211.0ms
(asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 9.1b

localize62.0ms

Local error

Found 4 expressions with local error:

8.2b
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))
0.6b
(asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))
0.2b
(pow (/ t l) 2)
0.2b
(/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 81
Calls

4 calls. Slowest were:

6.0ms
(/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))
3.0ms
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))
1.0ms
(pow (/ t l) 2)

series232.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

108.0ms
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))
86.0ms
(/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))
21.0ms
(asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))
17.0ms
(pow (/ t l) 2)

simplify6.9s

Counts
66 → 93
Calls

66 calls. Slowest were:

483.0ms
(- (* 1 1) (* (pow (/ Om Omc) 2) (pow (/ Om Omc) 2)))
483.0ms
(asin (sqrt (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1))))
449.0ms
(asin (sqrt (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1))))

prune1.5s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 9.0b

localize30.0ms

Local error

Found 4 expressions with local error:

8.2b
(sqrt (/ (/ (- 1 (pow (/ Om Omc) 2)) (sqrt (+ 1 (* 2 (pow (/ t l) 2))))) (sqrt (+ 1 (* 2 (pow (/ t l) 2))))))
8.2b
(sqrt (+ 1 (* 2 (pow (/ t l) 2))))
8.2b
(sqrt (+ 1 (* 2 (pow (/ t l) 2))))
0.6b
(asin (sqrt (/ (/ (- 1 (pow (/ Om Omc) 2)) (sqrt (+ 1 (* 2 (pow (/ t l) 2))))) (sqrt (+ 1 (* 2 (pow (/ t l) 2)))))))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 202
Calls

4 calls. Slowest were:

12.0ms
(sqrt (/ (/ (- 1 (pow (/ Om Omc) 2)) (sqrt (+ 1 (* 2 (pow (/ t l) 2))))) (sqrt (+ 1 (* 2 (pow (/ t l) 2))))))
1.0ms
(sqrt (+ 1 (* 2 (pow (/ t l) 2))))
1.0ms
(sqrt (+ 1 (* 2 (pow (/ t l) 2))))

series217.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

122.0ms
(sqrt (/ (/ (- 1 (pow (/ Om Omc) 2)) (sqrt (+ 1 (* 2 (pow (/ t l) 2))))) (sqrt (+ 1 (* 2 (pow (/ t l) 2))))))
40.0ms
(sqrt (+ 1 (* 2 (pow (/ t l) 2))))
32.0ms
(sqrt (+ 1 (* 2 (pow (/ t l) 2))))
23.0ms
(asin (sqrt (/ (/ (- 1 (pow (/ Om Omc) 2)) (sqrt (+ 1 (* 2 (pow (/ t l) 2))))) (sqrt (+ 1 (* 2 (pow (/ t l) 2)))))))

simplify22.7s

Counts
182 → 214
Calls

182 calls. Slowest were:

761.0ms
(asin (sqrt (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1))))
448.0ms
(asin (sqrt (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1))))
434.0ms
(asin (sqrt (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1))))

prune4.9s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 9.0b

localize44.0ms

Local error

Found 4 expressions with local error:

30.0b
(cbrt (pow (/ t l) 2))
30.0b
(cbrt (pow (/ t l) 2))
30.0b
(cbrt (pow (/ t l) 2))
8.2b
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (* (* (cbrt (pow (/ t l) 2)) (cbrt (pow (/ t l) 2))) (cbrt (pow (/ t l) 2)))))))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 74
Calls

4 calls. Slowest were:

5.0ms
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (* (* (cbrt (pow (/ t l) 2)) (cbrt (pow (/ t l) 2))) (cbrt (pow (/ t l) 2)))))))
1.0ms
(cbrt (pow (/ t l) 2))
1.0ms
(cbrt (pow (/ t l) 2))

series431.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

116.0ms
(cbrt (pow (/ t l) 2))
111.0ms
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (* (* (cbrt (pow (/ t l) 2)) (cbrt (pow (/ t l) 2))) (cbrt (pow (/ t l) 2)))))))
103.0ms
(cbrt (pow (/ t l) 2))
101.0ms
(cbrt (pow (/ t l) 2))

simplify4.4s

Counts
54 → 86
Calls

54 calls. Slowest were:

361.0ms
(exp (* 1/3 (- (* 2 (log (/ -1 l))) (* 2 (log (/ -1 t))))))
343.0ms
(exp (* 1/3 (- (* 2 (log (/ -1 l))) (* 2 (log (/ -1 t))))))
337.0ms
(exp (* 1/3 (- (* 2 (log (/ -1 l))) (* 2 (log (/ -1 t))))))

prune1.7s

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 9.0b

localize45.0ms

Local error

Found 4 expressions with local error:

30.0b
(cbrt (pow (/ t l) 2))
30.0b
(cbrt (pow (/ t l) 2))
16.5b
(cbrt (/ t l))
16.5b
(cbrt (/ t l))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 58
Calls

4 calls. Slowest were:

1.0ms
(cbrt (pow (/ t l) 2))
1.0ms
(cbrt (pow (/ t l) 2))
1.0ms
(cbrt (/ t l))

series368.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

101.0ms
(cbrt (/ t l))
97.0ms
(cbrt (pow (/ t l) 2))
85.0ms
(cbrt (pow (/ t l) 2))
85.0ms
(cbrt (/ t l))

simplify2.6s

Counts
38 → 70
Calls

38 calls. Slowest were:

461.0ms
(exp (* 1/3 (- (* 2 (log (/ -1 l))) (* 2 (log (/ -1 t))))))
401.0ms
(exp (* 1/3 (- (* 2 (log (/ -1 l))) (* 2 (log (/ -1 t))))))
236.0ms
(exp (* 1/3 (- (* 2 (log (/ 1 l))) (* 2 (log (/ 1 t))))))

prune1.5s

Pruning

10 alts after pruning (6 fresh and 4 done)

Merged error: 9.0b

regimes1.6s

Accuracy

0% (0.1b remaining)

Error of 10.0b against oracle of 10.0b and baseline of 10.0b

bsearch2.0ms

end0.0ms

sample6.3s

Algorithm
intervals