Average Error: 31.7 → 7.8
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.3613965471101462 \cdot 10^{+47} \lor \neg \left(t \le 1.1466904596145456 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \sin k\right) \cdot t\right)\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k + t \cdot \left(2 \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -1.3613965471101462e+47 or 1.1466904596145456e+69 < t

    1. Initial program 22.8

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    2. Using strategy rm

    3. Applied unpow322.8

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    4. Applied times-frac15.2

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    5. Applied associate-*l*14.0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    6. Using strategy rm

    7. Applied pow114.0

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \color{blue}{{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}^{1}}}\]

    8. Applied pow114.0

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \color{blue}{{\left(\tan k\right)}^{1}}\right) \cdot {\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}^{1}}\]

    9. Applied pow114.0

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}^{1}} \cdot {\left(\tan k\right)}^{1}\right) \cdot {\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}^{1}}\]

    10. Applied pow-prod-down14.0

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)}^{1}} \cdot {\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}^{1}}\]

    11. Applied pow-prod-down14.0

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}^{1}}}\]

    12. Simplified6.4

      \[\leadsto \frac{2}{{\color{blue}{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left(\sin k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)\right)}}^{1}}\]
    13. Using strategy rm

    14. Applied associate-*l*6.0

      \[\leadsto \frac{2}{{\color{blue}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\sin k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)\right)\right)}}^{1}}\]
    15. Using strategy rm

    16. Applied associate-*l*2.3

      \[\leadsto \frac{2}{{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\sin k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)\right)\right)}\right)}^{1}}\]

    if -1.3613965471101462e+47 < t < 1.1466904596145456e+69

    1. Initial program 41.7

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    2. Using strategy rm

    3. Applied unpow341.7

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    4. Applied times-frac35.6

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    5. Applied associate-*l*33.7

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    6. Using strategy rm

    7. Applied pow133.7

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \color{blue}{{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}^{1}}}\]

    8. Applied pow133.7

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \color{blue}{{\left(\tan k\right)}^{1}}\right) \cdot {\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}^{1}}\]

    9. Applied pow133.7

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}^{1}} \cdot {\left(\tan k\right)}^{1}\right) \cdot {\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}^{1}}\]

    10. Applied pow-prod-down33.7

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)}^{1}} \cdot {\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}^{1}}\]

    11. Applied pow-prod-down33.7

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}^{1}}}\]

    12. Simplified28.9

      \[\leadsto \frac{2}{{\color{blue}{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left(\sin k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)\right)}}^{1}}\]
    13. Using strategy rm

    14. Applied associate-*l*24.8

      \[\leadsto \frac{2}{{\color{blue}{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\sin k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)\right)\right)}}^{1}}\]

    15. Taylor expanded around inf 16.1

      \[\leadsto \frac{2}{{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(\frac{\sin k \cdot {k}^{2}}{\ell} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell}\right)}\right)}^{1}}\]

    16. Simplified14.1

      \[\leadsto \frac{2}{{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(\frac{\sin k}{\ell} \cdot \left(\left(t \cdot 2\right) \cdot t + k \cdot k\right)\right)}\right)}^{1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.3613965471101462 \cdot 10^{+47} \lor \neg \left(t \le 1.1466904596145456 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \sin k\right) \cdot t\right)\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k + t \cdot \left(2 \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))

Details

Time bar (total: 1.6m)Debug log

sample472.0ms

Algorithm
intervals

simplify149.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

149.0ms
(/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 31.9b

localize76.0ms

Local error

Found 4 expressions with local error:

16.1b
(/ (pow t 3) (* l l))
14.4b
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
2.9b
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
1.7b
(* (/ (pow t 3) (* l l)) (sin k))

rewrite107.0ms

Algorithm
rewrite-expression-head
Counts
4 → 120
Calls

4 calls. Slowest were:

65.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
25.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
11.0ms
(* (/ (pow t 3) (* l l)) (sin k))

series739.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

390.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
220.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
100.0ms
(* (/ (pow t 3) (* l l)) (sin k))
28.0ms
(/ (pow t 3) (* l l))

simplify16.1s

Counts
98 → 132
Calls

98 calls. Slowest were:

1.3s
(* (* l l) (- (+ 1 (pow (/ k t) 2)) 1))
736.0ms
(* (* (* l l) (cos k)) (- (+ 1 (pow (/ k t) 2)) 1))
671.0ms
(- (+ 1 (pow (/ k t) 2)) 1)

prune2.2s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 15.5b

localize21.0ms

Local error

Found 4 expressions with local error:

14.4b
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
8.0b
(/ (* t t) l)
2.9b
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
2.8b
(* (/ t l) (sin k))

rewrite189.0ms

Algorithm
rewrite-expression-head
Counts
4 → 115
Calls

4 calls. Slowest were:

134.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
44.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
5.0ms
(/ (* t t) l)

series673.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

381.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
211.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
64.0ms
(* (/ t l) (sin k))
16.0ms
(/ (* t t) l)

simplify21.7s

Counts
99 → 127
Calls

99 calls. Slowest were:

1.2s
(* (* l l) (- (+ 1 (pow (/ k t) 2)) 1))
948.0ms
(* (cos k) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))
732.0ms
(- (+ 1 (pow (/ k t) 2)) 1)

prune2.1s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 8.4b

localize39.0ms

Local error

Found 4 expressions with local error:

17.2b
(* (* (* (tan k) (/ t l)) (* (/ t l) t)) (* (sin k) (fma (/ k t) (/ k t) 2)))
2.8b
(* (tan k) (/ t l))
0.8b
(* (* (tan k) (/ t l)) (* (/ t l) t))
0.3b
(* (sin k) (fma (/ k t) (/ k t) 2))

rewrite109.0ms

Algorithm
rewrite-expression-head
Counts
4 → 110
Calls

4 calls. Slowest were:

78.0ms
(* (* (* (tan k) (/ t l)) (* (/ t l) t)) (* (sin k) (fma (/ k t) (/ k t) 2)))
18.0ms
(* (* (tan k) (/ t l)) (* (/ t l) t))
7.0ms
(* (tan k) (/ t l))

series499.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

205.0ms
(* (* (* (tan k) (/ t l)) (* (/ t l) t)) (* (sin k) (fma (/ k t) (/ k t) 2)))
124.0ms
(* (tan k) (/ t l))
120.0ms
(* (* (tan k) (/ t l)) (* (/ t l) t))
50.0ms
(* (sin k) (fma (/ k t) (/ k t) 2))

simplify13.0s

Counts
81 → 122
Calls

81 calls. Slowest were:

726.0ms
(* (* 2 (sin k)) (* (* (tan k) (/ t l)) (* (/ t l) t)))
635.0ms
(* (* (* (tan k) (/ t l)) (* t t)) (* (sin k) (fma (/ k t) (/ k t) 2)))
613.0ms
(* (* (* (sin k) t) (* (/ t l) t)) (* (sin k) (fma (/ k t) (/ k t) 2)))

prune2.1s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 7.7b

localize26.0ms

Local error

Found 4 expressions with local error:

14.3b
(* (* (/ t l) t) (* (sin k) (fma (/ k t) (/ k t) 2)))
7.0b
(* (* (tan k) (/ t l)) (* (* (/ t l) t) (* (sin k) (fma (/ k t) (/ k t) 2))))
2.8b
(* (tan k) (/ t l))
0.3b
(* (sin k) (fma (/ k t) (/ k t) 2))

rewrite94.0ms

Algorithm
rewrite-expression-head
Counts
4 → 112
Calls

4 calls. Slowest were:

58.0ms
(* (* (tan k) (/ t l)) (* (* (/ t l) t) (* (sin k) (fma (/ k t) (/ k t) 2))))
18.0ms
(* (* (/ t l) t) (* (sin k) (fma (/ k t) (/ k t) 2)))
8.0ms
(* (sin k) (fma (/ k t) (/ k t) 2))

series571.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

202.0ms
(* (* (/ t l) t) (* (sin k) (fma (/ k t) (/ k t) 2)))
201.0ms
(* (* (tan k) (/ t l)) (* (* (/ t l) t) (* (sin k) (fma (/ k t) (/ k t) 2))))
127.0ms
(* (tan k) (/ t l))
40.0ms
(* (sin k) (fma (/ k t) (/ k t) 2))

simplify15.8s

Counts
84 → 124
Calls

84 calls. Slowest were:

859.0ms
(* (* (* (/ t l) t) (* 2 (sin k))) (* (tan k) (/ t l)))
593.0ms
(* (* (tan k) t) (* (* t t) (* (sin k) (fma (/ k t) (/ k t) 2))))
593.0ms
(* (* (tan k) (/ t l)) (* (* (/ t l) t) (* 2 (sin k))))

prune2.0s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 6.4b

regimes387.0ms

Accuracy

74.8% (1.4b remaining)

Error of 7.8b against oracle of 6.4b and baseline of 12.0b

bsearch453.0ms

end0.0ms

sample15.7s

Algorithm
intervals