Average Error: 10.3 → 1.1
Time: 39.4s
Precision: 64
Internal Precision: 128
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le -4.0647357290577327 \cdot 10^{+34} \lor \neg \left(\frac{t}{\ell} \le 1.6327068826043366 \cdot 10^{+85}\right):\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \left|\frac{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right|\right))} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\left|\frac{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\sqrt{1 + {\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right| \cdot \left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right|\right))} - 1)^*\\ \end{array}\]

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. Split input into 2 regimes

  2. if (/ t l) < -4.0647357290577327e+34 or 1.6327068826043366e+85 < (/ t l)

    1. Initial program 23.4

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

    3. Applied expm1-log1p-u23.4

      \[\leadsto \color{blue}{(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right))} - 1)^*}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity23.4

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

    6. Applied add-cube-cbrt23.4

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{\color{blue}{\left(\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right) \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{1 \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right))} - 1)^*\]

    7. Applied times-frac23.4

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\sqrt{\color{blue}{\frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right))} - 1)^*\]

    8. Applied sqrt-prod23.4

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \color{blue}{\left(\sqrt{\frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1}} \cdot \sqrt{\frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)})} - 1)^*\]

    9. Simplified23.4

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\color{blue}{\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right|} \cdot \sqrt{\frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right))} - 1)^*\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt23.5

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \sqrt{\frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right))} - 1)^*\]

    12. Applied add-sqr-sqrt23.5

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \sqrt{\frac{\sqrt[3]{\color{blue}{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right))} - 1)^*\]

    13. Applied cbrt-prod23.5

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \sqrt{\frac{\color{blue}{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right))} - 1)^*\]

    14. Applied times-frac23.5

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \frac{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right))} - 1)^*\]

    15. Applied rem-sqrt-square23.5

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \color{blue}{\left|\frac{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right|}\right))} - 1)^*\]

    16. Taylor expanded around -inf 0.9

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \left|\frac{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\color{blue}{\frac{t \cdot \sqrt{2}}{\ell}}}\right|\right))} - 1)^*\]

    if -4.0647357290577327e+34 < (/ t l) < 1.6327068826043366e+85

    1. Initial program 1.1

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

    3. Applied expm1-log1p-u1.1

      \[\leadsto \color{blue}{(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right))} - 1)^*}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity1.1

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

    6. Applied add-cube-cbrt1.1

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{\color{blue}{\left(\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right) \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{1 \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right))} - 1)^*\]

    7. Applied times-frac1.1

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\sqrt{\color{blue}{\frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right))} - 1)^*\]

    8. Applied sqrt-prod1.1

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \color{blue}{\left(\sqrt{\frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1}} \cdot \sqrt{\frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)})} - 1)^*\]

    9. Simplified1.1

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\color{blue}{\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right|} \cdot \sqrt{\frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right))} - 1)^*\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt1.2

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \sqrt{\frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right))} - 1)^*\]

    12. Applied add-sqr-sqrt1.2

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \sqrt{\frac{\sqrt[3]{\color{blue}{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right))} - 1)^*\]

    13. Applied cbrt-prod1.2

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \sqrt{\frac{\color{blue}{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right))} - 1)^*\]

    14. Applied times-frac1.2

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}} \cdot \frac{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right))} - 1)^*\]

    15. Applied rem-sqrt-square1.2

      \[\leadsto (e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \color{blue}{\left|\frac{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right|}\right))} - 1)^*\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le -4.0647357290577327 \cdot 10^{+34} \lor \neg \left(\frac{t}{\ell} \le 1.6327068826043366 \cdot 10^{+85}\right):\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \left|\frac{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right|\right))} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\left|\frac{\sqrt[3]{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}{\sqrt{1 + {\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right| \cdot \left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right|\right))} - 1)^*\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(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: 37.3s)Debug log

sample268.0ms

Algorithm
intervals

simplify228.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

228.0ms
(asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 10.4b

localize40.0ms

Local error

Found 4 expressions with local error:

9.7b
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))
0.3b
(asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))
0.3b
(pow (/ t l) 2)
0.3b
(pow (/ Om Omc) 2)

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

4.0ms
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))
1.0ms
(pow (/ Om Omc) 2)
1.0ms
(pow (/ t l) 2)

series160.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

106.0ms
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))
20.0ms
(asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))
17.0ms
(pow (/ Om Omc) 2)
17.0ms
(pow (/ t l) 2)

simplify2.3s

Counts
42 → 84
Calls

42 calls. Slowest were:

209.0ms
(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)))))))
198.0ms
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ (pow 1 3) (pow (* 2 (pow (/ t l) 2)) 3))))
193.0ms
(sqrt (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))

prune1.5s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 10.4b

localize13.0ms

Local error

Found 4 expressions with local error:

9.7b
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))
0.3b
(asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))
0.3b
(pow (/ t l) 2)
0.3b
(pow (/ Om Omc) 2)

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

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

series161.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

111.0ms
(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))))))
17.0ms
(pow (/ Om Omc) 2)
16.0ms
(pow (/ t l) 2)

simplify2.3s

Counts
42 → 84
Calls

42 calls. Slowest were:

253.0ms
(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)))))))
248.0ms
(sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ (pow 1 3) (pow (* 2 (pow (/ t l) 2)) 3))))
216.0ms
(sqrt (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2))))))

prune1.2s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 10.4b

localize50.0ms

Local error

Found 4 expressions with local error:

9.7b
(sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (+ 1 (* 2 (pow (/ t l) 2)))))
0.3b
(asin (* (fabs (cbrt (- 1 (* (/ Om Omc) (/ Om Omc))))) (sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (+ 1 (* 2 (pow (/ t l) 2)))))))
0.3b
(pow (/ t l) 2)
0.3b
(* (/ Om Omc) (/ Om Omc))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 92
Calls

4 calls. Slowest were:

13.0ms
(sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (+ 1 (* 2 (pow (/ t l) 2)))))
7.0ms
(* (/ Om Omc) (/ Om Omc))
2.0ms
(pow (/ t l) 2)

series348.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

291.0ms
(sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (+ 1 (* 2 (pow (/ t l) 2)))))
28.0ms
(asin (* (fabs (cbrt (- 1 (* (/ Om Omc) (/ Om Omc))))) (sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (+ 1 (* 2 (pow (/ t l) 2)))))))
15.0ms
(pow (/ t l) 2)
14.0ms
(* (/ Om Omc) (/ Om Omc))

simplify3.4s

Counts
61 → 104
Calls

61 calls. Slowest were:

880.0ms
(* (* (* (/ Om Omc) (/ Om Omc)) (/ Om Omc)) (* (* (/ Om Omc) (/ Om Omc)) (/ Om Omc)))
169.0ms
(sqrt (/ (cbrt (- 1 (pow (/ Om Omc) 2))) (- (* 1 1) (* (* 2 (pow (/ t l) 2)) (* 2 (pow (/ t l) 2))))))
143.0ms
(asin (* (pow (- 1 (/ (pow Om 2) (pow Omc 2))) 1/6) (* (sqrt (/ 1 (+ (* 2 (/ (pow t 2) (pow l 2))) 1))) (fabs (pow (- 1 (/ (pow Om 2) (pow Omc 2))) 1/3)))))

prune1.6s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 10.4b

localize26.0ms

Local error

Found 4 expressions with local error:

9.6b
(sqrt (+ 1 (* 2 (pow (/ t l) 2))))
0.6b
(/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) (sqrt (+ 1 (* 2 (pow (/ t l) 2)))))
0.3b
(asin (* (fabs (cbrt (- 1 (* (/ Om Omc) (/ Om Omc))))) (fabs (/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) (sqrt (+ 1 (* 2 (pow (/ t l) 2))))))))
0.3b
(pow (/ t l) 2)

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 130
Calls

4 calls. Slowest were:

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

series280.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

217.0ms
(/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) (sqrt (+ 1 (* 2 (pow (/ t l) 2)))))
36.0ms
(sqrt (+ 1 (* 2 (pow (/ t l) 2))))
18.0ms
(asin (* (fabs (cbrt (- 1 (* (/ Om Omc) (/ Om Omc))))) (fabs (/ (cbrt (sqrt (- 1 (pow (/ Om Omc) 2)))) (sqrt (+ 1 (* 2 (pow (/ t l) 2))))))))
10.0ms
(pow (/ t l) 2)

simplify12.1s

Counts
152 → 142
Calls

152 calls. Slowest were:

809.0ms
(cbrt (sqrt (- (* 1 1) (* (pow (/ Om Omc) 2) (pow (/ Om Omc) 2)))))
295.0ms
(cbrt (sqrt (- (pow 1 3) (pow (pow (/ Om Omc) 2) 3))))
224.0ms
(asin (* (fabs (* (pow (- 1 (/ (pow Om 2) (pow Omc 2))) 1/6) (sqrt (/ 1 (+ (* 2 (/ (pow t 2) (pow l 2))) 1))))) (fabs (pow (- 1 (/ (pow Om 2) (pow Omc 2))) 1/3))))

prune3.5s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.9b

regimes2.6s

Accuracy

98.4% (0.2b remaining)

Error of 1.1b against oracle of 0.9b and baseline of 10.4b

bsearch14.0ms

end0.0ms

sample5.3s

Algorithm
intervals