Average Error: 33.4 → 27.7
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.158607358585534 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(\frac{\ell}{Om} \cdot \left(U \cdot \left(-2 \cdot n\right)\right)\right) \cdot (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n + \left(\ell \cdot 2\right))_* + \left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \mathbf{elif}\;t \le -1.4586901274772573 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)}\\ \mathbf{elif}\;t \le 4.9744455505630306 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\sqrt{\left(t - (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}} \cdot \sqrt{\sqrt{\left(t - (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}}\\ \mathbf{elif}\;t \le 8.69812872684918 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

  1. Split input into 4 regimes

  2. if t < -7.158607358585534e+32

    1. Initial program 33.7

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm

    3. Applied associate-/l*31.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    4. Using strategy rm

    5. Applied unpow231.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(U - U*\right)\right)}\]

    6. Applied associate-*r*31.1

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(U - U*\right)\right)}\]
    7. Using strategy rm

    8. Applied pow131.1

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)}^{1}}}\]

    9. Applied pow131.1

      \[\leadsto \sqrt{\color{blue}{{\left(\left(2 \cdot n\right) \cdot U\right)}^{1}} \cdot {\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)}^{1}}\]

    10. Applied pow-prod-down31.1

      \[\leadsto \sqrt{\color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}^{1}}}\]

    11. Simplified31.1

      \[\leadsto \sqrt{{\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t - (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*\right)\right)}}^{1}}\]
    12. Using strategy rm

    13. Applied sub-neg31.1

      \[\leadsto \sqrt{{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \color{blue}{\left(t + \left(-(\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*\right)\right)}\right)}^{1}}\]

    14. Applied distribute-rgt-in31.1

      \[\leadsto \sqrt{{\color{blue}{\left(t \cdot \left(2 \cdot \left(U \cdot n\right)\right) + \left(-(\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)\right)}}^{1}}\]

    15. Simplified29.3

      \[\leadsto \sqrt{{\left(t \cdot \left(2 \cdot \left(U \cdot n\right)\right) + \color{blue}{\left(\left(\left(-2 \cdot n\right) \cdot U\right) \cdot \frac{\ell}{Om}\right) \cdot (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n + \left(2 \cdot \ell\right))_*}\right)}^{1}}\]

    if -7.158607358585534e+32 < t < -1.4586901274772573e-230 or 4.9744455505630306e-176 < t < 8.69812872684918e+198

    1. Initial program 31.4

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm

    3. Applied associate-/l*28.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    4. Using strategy rm

    5. Applied unpow228.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(U - U*\right)\right)}\]

    6. Applied associate-*r*27.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(U - U*\right)\right)}\]
    7. Using strategy rm

    8. Applied associate-*l*26.8

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}}\]

    if -1.4586901274772573e-230 < t < 4.9744455505630306e-176

    1. Initial program 36.3

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm

    3. Applied associate-/l*33.5

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    4. Using strategy rm

    5. Applied unpow233.5

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(U - U*\right)\right)}\]

    6. Applied associate-*r*31.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(U - U*\right)\right)}\]
    7. Using strategy rm

    8. Applied pow131.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)}^{1}}}\]

    9. Applied pow131.7

      \[\leadsto \sqrt{\color{blue}{{\left(\left(2 \cdot n\right) \cdot U\right)}^{1}} \cdot {\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)}^{1}}\]

    10. Applied pow-prod-down31.7

      \[\leadsto \sqrt{\color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}^{1}}}\]

    11. Simplified30.7

      \[\leadsto \sqrt{{\color{blue}{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t - (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*\right)\right)}}^{1}}\]
    12. Using strategy rm

    13. Applied add-sqr-sqrt30.9

      \[\leadsto \color{blue}{\sqrt{\sqrt{{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t - (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*\right)\right)}^{1}}} \cdot \sqrt{\sqrt{{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(t - (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*\right)\right)}^{1}}}}\]

    if 8.69812872684918e+198 < t

    1. Initial program 39.2

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm

    3. Applied associate-/l*36.8

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    4. Using strategy rm

    5. Applied unpow236.8

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(U - U*\right)\right)}\]

    6. Applied associate-*r*36.3

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(U - U*\right)\right)}\]
    7. Using strategy rm

    8. Applied sqrt-prod23.1

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification27.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.158607358585534 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{\left(\frac{\ell}{Om} \cdot \left(U \cdot \left(-2 \cdot n\right)\right)\right) \cdot (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n + \left(\ell \cdot 2\right))_* + \left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \mathbf{elif}\;t \le -1.4586901274772573 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)}\\ \mathbf{elif}\;t \le 4.9744455505630306 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\sqrt{\left(t - (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}} \cdot \sqrt{\sqrt{\left(t - (\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_*\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}}\\ \mathbf{elif}\;t \le 8.69812872684918 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))

Details

Time bar (total: 1.3m)Debug log

sample1.2s

Algorithm
intervals
Results
761×(pre true 80)
464×(body nan 80)
297×(body real 80)

simplify150.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
149.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

prune21.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 32.7b

localize58.0ms

Local error

Found 4 expressions with local error:

17.9b
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
10.1b
(* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))
9.8b
(/ (* l l) Om)
7.3b
(* (* n (pow (/ l Om) 2)) (- U U*))

rewrite61.0ms

Algorithm
rewrite-expression-head
Rules
22×associate-*r/
11×flip--
11×flip3--
9×distribute-lft-in
9×distribute-rgt-in
8×add-cube-cbrt
8×frac-sub
8×add-exp-log
8×add-cbrt-cube
8×*-un-lft-identity
8×pow1
8×add-sqr-sqrt
6×prod-diff
6×associate-*r*
6×sqrt-div
6×sub-neg
4×add-log-exp
4×log1p-expm1-u
4×associate--l+
4×expm1-log1p-u
2×cbrt-unprod
2×*-commutative
2×prod-exp
2×pow-prod-down
2×associate-*l*
1×associate-/l*
1×div-inv
1×sqrt-prod
1×pow1/2
1×frac-2neg
1×clear-num
Counts
4 → 90
Calls
4 calls:
Slowest
23.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))
18.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
11.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))
4.0ms
(/ (* l l) Om)

series1.9s

Counts
4 → 12
Calls
4 calls:
Slowest
1.5s
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
298.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))
84.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))
10.0ms
(/ (* l l) Om)

simplify10.2s

Counts
58 → 102
Calls
58 calls:
Slowest
589.0ms
(* (fma (- (- U U*)) (* n (pow (/ l Om) 2)) (* (- U U*) (* n (pow (/ l Om) 2)))) (* (* 2 n) U))
512.0ms
(* (fma (- (- U U*)) (* n (pow (/ l Om) 2)) (* (- U U*) (* n (pow (/ l Om) 2)))) (* (* 2 n) U))
500.0ms
(* (fma (- (- U U*)) (* n (pow (/ l Om) 2)) (* (- U U*) (* n (pow (/ l Om) 2)))) (* (* 2 n) U))
424.0ms
(sqrt (* (* (* 2 n) U) (- (* (- (pow t 3) (pow (* 2 (/ (* l l) Om)) 3)) (+ U U*)) (* (+ (* t t) (+ (* (* 2 (/ (* l l) Om)) (* 2 (/ (* l l) Om))) (* t (* 2 (/ (* l l) Om))))) (* (* n (pow (/ l Om) 2)) (- (* U U) (* U* U*)))))))
382.0ms
(+ (log (* (* 2 n) U)) (log (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

prune2.0s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 19.9b

localize18.0ms

Local error

Found 4 expressions with local error:

17.9b
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
10.1b
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
7.3b
(* (* n (pow (/ l Om) 2)) (- U U*))
5.2b
(* n (pow (/ l Om) 2))

rewrite52.0ms

Algorithm
rewrite-expression-head
Rules
22×associate-*r/
14×associate-*r*
11×flip--
11×flip3--
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
9×distribute-lft-in
9×distribute-rgt-in
8×frac-sub
8×add-exp-log
8×add-cbrt-cube
8×pow1
6×prod-diff
6×sqrt-div
6×sub-neg
4×add-log-exp
4×log1p-expm1-u
4×associate--l+
4×unpow-prod-down
4×expm1-log1p-u
3×*-commutative
2×cbrt-unprod
2×prod-exp
2×pow-prod-down
2×associate-*l*
1×unpow2
1×div-inv
1×sqrt-prod
1×pow1/2
Counts
4 → 95
Calls
4 calls:
Slowest
28.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
15.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
6.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))
2.0ms
(* n (pow (/ l Om) 2))

series1.5s

Counts
4 → 12
Calls
4 calls:
Slowest
1.1s
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
306.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
81.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))
35.0ms
(* n (pow (/ l Om) 2))

simplify11.3s

Counts
66 → 107
Calls
66 calls:
Slowest
985.0ms
(* (fma (- (- U U*)) (* n (pow (/ l Om) 2)) (* (- U U*) (* n (pow (/ l Om) 2)))) (* (* 2 n) U))
594.0ms
(* (fma (- (- U U*)) (* n (pow (/ l Om) 2)) (* (- U U*) (* n (pow (/ l Om) 2)))) (* (* 2 n) U))
543.0ms
(* (fma (- (- U U*)) (* n (pow (/ l Om) 2)) (* (- U U*) (* n (pow (/ l Om) 2)))) (* (* 2 n) U))
411.0ms
(cbrt (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
402.0ms
(+ (log (* (* 2 n) U)) (log (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

prune2.1s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 18.9b

localize32.0ms

Local error

Found 4 expressions with local error:

17.9b
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* (* n (/ l Om)) (/ l Om)) (- U U*)))))
10.1b
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* (* n (/ l Om)) (/ l Om)) (- U U*))))
7.3b
(* (* (* n (/ l Om)) (/ l Om)) (- U U*))
4.4b
(* n (/ l Om))

rewrite104.0ms

Algorithm
rewrite-expression-head
Rules
104×associate-*r/
45×frac-times
44×flip--
44×frac-sub
44×flip3--
30×associate-*l/
24×sqrt-div
11×add-exp-log
11×add-cbrt-cube
11×pow1
10×associate-*r*
9×add-cube-cbrt
9×distribute-lft-in
9×*-un-lft-identity
9×add-sqr-sqrt
9×distribute-rgt-in
6×prod-diff
6×sub-neg
4×add-log-exp
4×log1p-expm1-u
4×cbrt-unprod
4×associate--l+
4×prod-exp
4×pow-prod-down
4×expm1-log1p-u
3×*-commutative
2×associate-*l*
1×div-inv
1×sqrt-prod
1×pow1/2
Counts
4 → 140
Calls
4 calls:
Slowest
55.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* (* n (/ l Om)) (/ l Om)) (- U U*))))
32.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* (* n (/ l Om)) (/ l Om)) (- U U*)))))
13.0ms
(* (* (* n (/ l Om)) (/ l Om)) (- U U*))
1.0ms
(* n (/ l Om))

series1.6s

Counts
4 → 12
Calls
4 calls:
Slowest
1.3s
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* (* n (/ l Om)) (/ l Om)) (- U U*)))))
300.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* (* n (/ l Om)) (/ l Om)) (- U U*))))
76.0ms
(* (* (* n (/ l Om)) (/ l Om)) (- U U*))
13.0ms
(* n (/ l Om))

simplify22.3s

Counts
117 → 152
Calls
117 calls:
Slowest
541.0ms
(* (+ t (* 2 (/ l (/ Om l)))) (* Om (+ (* U U) (+ (* U* U*) (* U U*)))))
540.0ms
(sqrt (* (* (* 2 n) U) (- (* (- (* t t) (* (* 2 (/ l (/ Om l))) (* 2 (/ l (/ Om l))))) (* Om Om)) (* (+ t (* 2 (/ l (/ Om l)))) (* (* (* n l) l) (- U U*))))))
514.0ms
(* (+ t (* 2 (/ l (/ Om l)))) (* Om (+ (* U U) (+ (* U* U*) (* U U*)))))
501.0ms
(sqrt (* (* (* 2 n) U) (- (* (- (pow t 3) (pow (* 2 (/ l (/ Om l))) 3)) (* Om (+ (* U U) (+ (* U* U*) (* U U*))))) (* (+ (* t t) (+ (* (* 2 (/ l (/ Om l))) (* 2 (/ l (/ Om l)))) (* t (* 2 (/ l (/ Om l)))))) (* (* (* n l) (/ l Om)) (- (pow U 3) (pow U* 3)))))))
461.0ms
(* (* (* n l) (/ l Om)) (- (* U U) (* U* U*)))

prune3.3s

Pruning

18 alts after pruning (18 fresh and 0 done)

Merged error: 16.6b

localize62.0ms

Local error

Found 4 expressions with local error:

17.9b
(sqrt (pow (* (* 2 (* U n)) (- t (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om))))) 1))
10.1b
(* (* 2 (* U n)) (- t (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om)))))
5.9b
(* (/ l Om) (- U U*))
4.7b
(fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om)))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
9×add-sqr-sqrt
8×add-cube-cbrt
8×add-exp-log
8×add-cbrt-cube
8×*-un-lft-identity
8×pow1
6×associate-*r*
5×associate-*l*
4×add-log-exp
4×log1p-expm1-u
4×sqrt-prod
4×associate-*r/
4×distribute-lft-in
4×sub-neg
4×expm1-log1p-u
4×distribute-rgt-in
3×flip--
3×flip3--
2×cbrt-unprod
2×*-commutative
2×prod-exp
2×pow-prod-down
2×frac-times
1×div-inv
1×fma-udef
1×associate-*l/
1×pow1/2
1×unpow-prod-down
1×rem-sqrt-square
Counts
4 → 77
Calls
4 calls:
Slowest
10.0ms
(* (* 2 (* U n)) (- t (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om)))))
7.0ms
(* (/ l Om) (- U U*))
5.0ms
(sqrt (pow (* (* 2 (* U n)) (- t (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om))))) 1))
0.0ms
(fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om)))

series1.9s

Counts
4 → 12
Calls
4 calls:
Slowest
1.5s
(sqrt (pow (* (* 2 (* U n)) (- t (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om))))) 1))
290.0ms
(* (* 2 (* U n)) (- t (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om)))))
85.0ms
(fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om)))
48.0ms
(* (/ l Om) (- U U*))

simplify5.9s

Counts
48 → 89
Calls
48 calls:
Slowest
731.0ms
(* (* 2 (* U n)) (- t (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om)))))
476.0ms
(+ (* t t) (+ (* (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om))) (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om)))) (* t (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om))))))
319.0ms
(- (+ (* 2 (/ (pow l 2) Om)) (/ (* U (* n (pow l 2))) (pow Om 2))) (/ (* n (* U* (pow l 2))) (pow Om 2)))
302.0ms
(sqrt (* (cbrt (pow (* (* 2 (* U n)) (- t (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om))))) 1)) (cbrt (pow (* (* 2 (* U n)) (- t (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om))))) 1))))
276.0ms
(* (- (fma (* (/ l Om) (- U U*)) (* n (/ l Om)) (* (* 2 l) (/ l Om)))) (* 2 (* U n)))

prune1.6s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 15.4b

regimes1.3s

Accuracy

5.3% (9.8b remaining)

Error of 27.7b against oracle of 18.0b and baseline of 28.3b

bsearch1.3s

end0.0ms

sample7.4s

Algorithm
intervals
Results
23967×(pre true 80)
14776×(body nan 80)
9191×(body real 80)