Average Error: 33.4 → 29.4
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.3577287425079678 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 3.2980276914506116 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\\ \end{array}\]

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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -2.3577287425079678e+52

    1. Initial program 34.2

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

    3. Applied associate-/l*32.1

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

    5. Applied associate-*l*32.6

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

    7. Applied unpow232.6

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

    8. Applied associate-*l*32.4

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

    if -2.3577287425079678e+52 < t < 3.2980276914506116e+204

    1. Initial program 32.5

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

    3. Applied associate-/l*29.7

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

    5. Applied associate-*l*30.4

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

    7. Applied unpow230.4

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

    8. Applied associate-*l*29.4

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

    10. Applied associate-*l*29.2

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

    if 3.2980276914506116e+204 < 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 sqrt-prod23.2

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

  4. Final simplification29.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.3577287425079678 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 3.2980276914506116 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 
(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

sample855.0ms

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

simplify124.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
124.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

prune22.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 32.7b

localize53.0ms

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*))

rewrite40.0ms

Algorithm
rewrite-expression-head
Rules
22×associate-*r/
11×flip--
11×flip3--
8×frac-sub
8×add-exp-log
8×add-cbrt-cube
8×pow1
6×add-cube-cbrt
6×associate-*r*
6×sqrt-div
6×distribute-lft-in
6×*-un-lft-identity
6×sub-neg
6×add-sqr-sqrt
6×distribute-rgt-in
4×add-log-exp
4×associate--l+
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 → 76
Calls
4 calls:
Slowest
19.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))
10.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
6.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))
2.0ms
(/ (* l l) Om)

series1.7s

Counts
4 → 12
Calls
4 calls:
Slowest
1.3s
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
334.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))
72.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))
15.0ms
(/ (* l l) Om)

simplify8.5s

Counts
52 → 88
Calls
52 calls:
Slowest
632.0ms
(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*)))))))
371.0ms
(* (+ t (* 2 (/ (* l l) Om))) (+ U U*))
351.0ms
(* (- U*) (* n (pow (/ l Om) 2)))
344.0ms
(* (* n (pow (/ l Om) 2)) (- U*))
343.0ms
(+ (log (* (* 2 n) U)) (log (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

prune1.7s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 20.2b

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))

rewrite54.0ms

Algorithm
rewrite-expression-head
Rules
22×associate-*r/
14×associate-*r*
11×flip--
11×flip3--
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×sqrt-div
6×distribute-lft-in
6×sub-neg
6×distribute-rgt-in
4×add-log-exp
4×associate--l+
4×unpow-prod-down
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 → 81
Calls
4 calls:
Slowest
37.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
8.0ms
(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
1.2s
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
249.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
89.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))
50.0ms
(* n (pow (/ l Om) 2))

simplify8.1s

Counts
60 → 93
Calls
60 calls:
Slowest
389.0ms
(* (+ t (* 2 (/ l (/ Om l)))) (+ U U*))
368.0ms
(+ (log (* n (pow (/ l Om) 2))) (log (- U U*)))
315.0ms
(+ (log (* (* 2 n) U)) (log (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
308.0ms
(* (- U*) (* n (pow (/ l Om) 2)))
308.0ms
(* (+ t (* 2 (/ l (/ Om l)))) (+ (* U U) (+ (* U* U*) (* U U*))))

prune1.7s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 19.2b

localize26.0ms

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*)))))
6.0b
(* n (* (pow (/ l Om) 2) (- U U*)))
5.6b
(* (pow (/ l Om) 2) (- U U*))

rewrite64.0ms

Algorithm
rewrite-expression-head
Rules
34×associate-*r/
14×distribute-lft-in
14×distribute-rgt-in
12×flip--
12×flip3--
11×add-exp-log
10×sub-neg
9×associate-*l*
8×add-cube-cbrt
8×frac-sub
8×add-cbrt-cube
8×*-un-lft-identity
8×add-sqr-sqrt
7×associate-*r*
6×sqrt-div
6×pow1
4×add-log-exp
4×associate--l+
4×prod-exp
4×unpow-prod-down
3×*-commutative
2×cbrt-unprod
1×unpow2
1×div-inv
1×sqrt-prod
1×pow-exp
1×pow-prod-down
1×pow1/2
1×pow-to-exp
Counts
4 → 92
Calls
4 calls:
Slowest
31.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* n (* (pow (/ l Om) 2) (- U U*)))))
22.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* n (* (pow (/ l Om) 2) (- U U*))))))
4.0ms
(* n (* (pow (/ l Om) 2) (- U U*)))
4.0ms
(* (pow (/ l Om) 2) (- U U*))

series1.6s

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*))))))
369.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* n (* (pow (/ l Om) 2) (- U U*)))))
108.0ms
(* n (* (pow (/ l Om) 2) (- U U*)))
54.0ms
(* (pow (/ l Om) 2) (- U U*))

simplify10.5s

Counts
71 → 104
Calls
71 calls:
Slowest
683.0ms
(* (* (* (pow (/ l Om) 2) (pow (/ l Om) 2)) (pow (/ l Om) 2)) (* (* (- U U*) (- U U*)) (- U U*)))
661.0ms
(sqrt (* (* (* 2 n) U) (- (* (- (pow t 3) (pow (* 2 (/ l (/ Om l))) 3)) (+ U U*)) (* (+ (* t t) (+ (* (* 2 (/ l (/ Om l))) (* 2 (/ l (/ Om l)))) (* t (* 2 (/ l (/ Om l)))))) (* n (* (pow (/ l Om) 2) (- (* U U) (* U* U*))))))))
351.0ms
(* n (* (- U*) (pow (/ l Om) 2)))
328.0ms
(sqrt (* (* (* 2 n) U) (- (* (- (pow t 3) (pow (* 2 (/ l (/ Om l))) 3)) (+ (* U U) (+ (* U* U*) (* U U*)))) (* (+ (* t t) (+ (* (* 2 (/ l (/ Om l))) (* 2 (/ l (/ Om l)))) (* t (* 2 (/ l (/ Om l)))))) (* n (* (pow (/ l Om) 2) (- (pow U 3) (pow U* 3))))))))
325.0ms
(* n (* (- U*) (pow (/ l Om) 2)))

prune1.9s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 19.2b

localize33.0ms

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*))))))
6.0b
(* n (* (/ l Om) (* (/ l Om) (- U U*))))
5.9b
(* (/ l Om) (- U U*))

rewrite200.0ms

Algorithm
rewrite-expression-head
Rules
150×associate-*r/
47×frac-times
46×flip--
46×flip3--
44×frac-sub
30×distribute-lft-in
30×distribute-rgt-in
24×sqrt-div
16×associate-*l/
14×sub-neg
8×add-exp-log
8×add-cbrt-cube
8×pow1
7×add-cube-cbrt
7×associate-*r*
7×*-un-lft-identity
7×add-sqr-sqrt
5×associate-*l*
4×add-log-exp
4×associate--l+
3×*-commutative
2×cbrt-unprod
2×prod-exp
2×pow-prod-down
1×div-inv
1×sqrt-prod
1×pow1/2
Counts
4 → 143
Calls
4 calls:
Slowest
92.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* n (* (/ l Om) (* (/ l Om) (- U U*))))))
69.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* n (* (/ l Om) (* (/ l Om) (- U U*)))))))
25.0ms
(* n (* (/ l Om) (* (/ l Om) (- U U*))))
7.0ms
(* (/ l Om) (- U U*))

series1.6s

Counts
4 → 12
Calls
4 calls:
Slowest
1.2s
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* n (* (/ l Om) (* (/ l Om) (- U U*)))))))
311.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* n (* (/ l Om) (* (/ l Om) (- U U*))))))
94.0ms
(* n (* (/ l Om) (* (/ l Om) (- U U*))))
57.0ms
(* (/ l Om) (- U U*))

simplify20.5s

Counts
121 → 155
Calls
121 calls:
Slowest
771.0ms
(* (* (* (/ l Om) (/ l Om)) (/ l Om)) (* (* (- U U*) (- U U*)) (- U U*)))
658.0ms
(sqrt (* (* (* 2 n) U) (- (* (- (* t t) (* (* 2 (/ l (/ Om l))) (* 2 (/ l (/ Om l))))) (* Om (+ U U*))) (* (+ t (* 2 (/ l (/ Om l)))) (* n (* (/ l Om) (* l (- (* U U) (* U* U*)))))))))
480.0ms
(* (+ (* t t) (+ (* (* 2 (/ l (/ Om l))) (* 2 (/ l (/ Om l)))) (* t (* 2 (/ l (/ Om l)))))) (* Om Om))
419.0ms
(* (+ t (* 2 (/ l (/ Om l)))) (* Om Om))
406.0ms
(* (+ (* t t) (+ (* (* 2 (/ l (/ Om l))) (* 2 (/ l (/ Om l)))) (* t (* 2 (/ l (/ Om l)))))) Om)

prune3.0s

Pruning

18 alts after pruning (17 fresh and 1 done)

Merged error: 17.4b

regimes1.6s

Accuracy

12.6% (9.5b remaining)

Error of 29.4b against oracle of 19.9b and baseline of 30.8b

bsearch1.6s

end0.0ms

sample8.1s

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