Average Error: 10.0 → 10.0
Time: 46.9s
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 2019002 
(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: 45.4s)Debug log

sample163.0ms

Algorithm
intervals

simplify200.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

199.0ms
(asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 10.1b

localize47.0ms

Local error

Found 4 expressions with local error:

9.3b
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))
0.7b
(asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))
0.2b
(pow (/ Om Omc) 2)
0.2b
(pow (/ t l) 2)

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

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

series133.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

96.0ms
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))
17.0ms
(asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))
10.0ms
(pow (/ Om Omc) 2)
10.0ms
(pow (/ t l) 2)

simplify3.3s

Counts
42 → 76
Calls

42 calls. Slowest were:

558.0ms
(sqrt (* (cbrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))) (cbrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))
450.0ms
(asin (sqrt (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1))))
347.0ms
(asin (sqrt (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1))))

prune1.2s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 10.0b

localize46.0ms

Local error

Found 4 expressions with local error:

15.0b
(cbrt (+ (* (/ t l) (* (/ t l) 2)) 1))
15.0b
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))
2.2b
(sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (cbrt (+ 1 (* 2 (pow (/ t l) 2))))))
0.7b
(asin (* (fabs (/ (cbrt (- 1 (* (/ Om Omc) (/ Om Omc)))) (cbrt (+ (* (/ t l) (* (/ t l) 2)) 1)))) (sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (cbrt (+ 1 (* 2 (pow (/ t l) 2))))))))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

7.0ms
(sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (cbrt (+ 1 (* 2 (pow (/ t l) 2))))))
6.0ms
(cbrt (+ (* (/ t l) (* (/ t l) 2)) 1))
1.0ms
(asin (* (fabs (/ (cbrt (- 1 (* (/ Om Omc) (/ Om Omc)))) (cbrt (+ (* (/ t l) (* (/ t l) 2)) 1)))) (sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (cbrt (+ 1 (* 2 (pow (/ t l) 2))))))))

series501.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

203.0ms
(sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (cbrt (+ 1 (* 2 (pow (/ t l) 2))))))
141.0ms
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))
121.0ms
(cbrt (+ (* (/ t l) (* (/ t l) 2)) 1))
36.0ms
(asin (* (fabs (/ (cbrt (- 1 (* (/ Om Omc) (/ Om Omc)))) (cbrt (+ (* (/ t l) (* (/ t l) 2)) 1)))) (sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (cbrt (+ 1 (* 2 (pow (/ t l) 2))))))))

simplify8.1s

Counts
70 → 102
Calls

70 calls. Slowest were:

495.0ms
(asin (* (pow (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1)) 1/6) (fabs (pow (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1)) 1/3))))
452.0ms
(cbrt (+ (pow (* (/ t l) (* (/ t l) 2)) 3) (pow 1 3)))
403.0ms
(asin (* (pow (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1)) 1/6) (fabs (pow (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1)) 1/3))))

prune2.7s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 10.0b

localize57.0ms

Local error

Found 4 expressions with local error:

15.0b
(cbrt (+ (* (/ t l) (* (/ t l) 2)) 1))
15.0b
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))
2.2b
(sqrt (/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) (cbrt (+ 1 (* 2 (pow (/ t l) 2))))))
0.7b
(asin (* (fabs (/ (cbrt (- 1 (* (/ Om Omc) (/ Om Omc)))) (cbrt (+ (* (/ t l) (* (/ t l) 2)) 1)))) (* (sqrt (/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) 1)) (sqrt (/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) (cbrt (+ 1 (* 2 (pow (/ t l) 2)))))))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 110
Calls

4 calls. Slowest were:

9.0ms
(cbrt (+ (* (/ t l) (* (/ t l) 2)) 1))
8.0ms
(sqrt (/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) (cbrt (+ 1 (* 2 (pow (/ t l) 2))))))
2.0ms
(asin (* (fabs (/ (cbrt (- 1 (* (/ Om Omc) (/ Om Omc)))) (cbrt (+ (* (/ t l) (* (/ t l) 2)) 1)))) (* (sqrt (/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) 1)) (sqrt (/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) (cbrt (+ 1 (* 2 (pow (/ t l) 2)))))))))

series602.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

288.0ms
(sqrt (/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) (cbrt (+ 1 (* 2 (pow (/ t l) 2))))))
144.0ms
(cbrt (+ (* (/ t l) (* (/ t l) 2)) 1))
137.0ms
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))
33.0ms
(asin (* (fabs (/ (cbrt (- 1 (* (/ Om Omc) (/ Om Omc)))) (cbrt (+ (* (/ t l) (* (/ t l) 2)) 1)))) (* (sqrt (/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) 1)) (sqrt (/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) (cbrt (+ 1 (* 2 (pow (/ t l) 2)))))))))

simplify10.1s

Counts
90 → 122
Calls

90 calls. Slowest were:

504.0ms
(asin (* (pow (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1)) 1/6) (fabs (pow (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1)) 1/3))))
500.0ms
(asin (* (pow (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1)) 1/6) (fabs (pow (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1)) 1/3))))
482.0ms
(asin (* (pow (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1)) 1/6) (fabs (pow (/ (- 1 (/ (pow Om 2) (pow Omc 2))) (+ (* 2 (/ (pow t 2) (pow l 2))) 1)) 1/3))))

prune3.8s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 10.0b

localize46.0ms

Local error

Found 4 expressions with local error:

15.0b
(cbrt (+ (* (/ t l) (* (/ t l) 2)) 1))
15.0b
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))
15.0b
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))
15.0b
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

4.0ms
(cbrt (+ (* (/ t l) (* (/ t l) 2)) 1))
1.0ms
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))
1.0ms
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))

series575.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

159.0ms
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))
149.0ms
(cbrt (+ (* (/ t l) (* (/ t l) 2)) 1))
145.0ms
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))
123.0ms
(cbrt (+ 1 (* 2 (pow (/ t l) 2))))

simplify4.5s

Counts
32 → 64
Calls

32 calls. Slowest were:

527.0ms
(cbrt (- (* 1 1) (* (* 2 (pow (/ t l) 2)) (* 2 (pow (/ t l) 2)))))
441.0ms
(cbrt (+ (pow (* (/ t l) (* (/ t l) 2)) 3) (pow 1 3)))
295.0ms
(exp (* 1/3 (- (+ (log 2) (* 2 (log (/ -1 l)))) (* 2 (log (/ -1 t))))))

prune2.0s

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 10.0b

regimes1.4s

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample5.8s

Algorithm
intervals