Average Error: 31.4 → 12.7
Time: 1.7m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sin k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right) \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{\ell}{t}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 31.4

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

  2. Initial simplification24.5

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

  4. Applied add-sqr-sqrt24.5

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

  5. Applied times-frac18.3

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

  6. Applied times-frac16.5

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

  8. Applied add-sqr-sqrt16.4

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

  9. Applied *-un-lft-identity16.4

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

  10. Applied times-frac13.6

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

  11. Applied times-frac12.7

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

  12. Simplified12.7

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

  14. Applied add-sqr-sqrt12.7

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

  15. Applied *-un-lft-identity12.7

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

  16. Applied add-cube-cbrt12.9

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}\right) \cdot \sqrt[3]{\frac{2}{t}}}}{1 \cdot \sin k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right)\]

  17. Applied times-frac12.9

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sin k}}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}} \cdot \sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right)\]

  18. Applied times-frac12.7

    \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{1}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sin k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right)} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right)\]

  19. Simplified12.7

    \[\leadsto \left(\color{blue}{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sin k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right) \cdot \left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right)\]

  20. Final simplification12.7

    \[\leadsto \left(\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sin k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right) \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{\ell}{t}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right)\]

Reproduce

herbie shell --seed 2018362 +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: 1.3m)Debug log

start757.0ms

Algorithm
intervals

setup162.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 21.3b

localize42.0ms

Local error

Found 4 expressions with local error:

13.7b
(/ (/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k))) (fma (/ k t) (/ k t) 2))
10.8b
(/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k)))
0.8b
(* (/ 2 t) (* (/ l t) (/ l t)))
0.2b
(* (/ l t) (/ l t))

rewrite64.0ms

Algorithm
rewrite-expression-head
Counts
4 → 144
Calls

4 calls. Slowest were:

25.0ms
(/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k)))
24.0ms
(/ (/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k))) (fma (/ k t) (/ k t) 2))
9.0ms
(* (/ 2 t) (* (/ l t) (/ l t)))

series566.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

264.0ms
(/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k)))
242.0ms
(/ (/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k))) (fma (/ k t) (/ k t) 2))
37.0ms
(* (/ 2 t) (* (/ l t) (/ l t)))
23.0ms
(* (/ l t) (/ l t))

simplify24.5s

Counts
127 → 156
Calls

127 calls. Slowest were:

970.0ms
(* (* (* (/ l t) (/ l t)) (/ l t)) (* (* (/ l t) (/ l t)) (/ l t)))
929.0ms
(/ (/ (* (* (* (/ 2 t) (/ 2 t)) (/ 2 t)) (* (* (* (/ l t) (/ l t)) (/ l t)) (* (* (/ l t) (/ l t)) (/ l t)))) (* (* (* (sin k) (sin k)) (sin k)) (* (* (tan k) (tan k)) (tan k)))) (* (* (fma (/ k t) (/ k t) 2) (fma (/ k t) (/ k t) 2)) (fma (/ k t) (/ k t) 2)))
847.0ms
(/ (/ (* (* (* (/ 2 t) (/ 2 t)) (/ 2 t)) (* (* (* (/ l t) (/ l t)) (* (/ l t) (/ l t))) (* (/ l t) (/ l t)))) (* (* (* (sin k) (sin k)) (sin k)) (* (* (tan k) (tan k)) (tan k)))) (* (* (fma (/ k t) (/ k t) 2) (fma (/ k t) (/ k t) 2)) (fma (/ k t) (/ k t) 2)))

prune2.8s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 10.7b

localize41.0ms

Local error

Found 4 expressions with local error:

11.3b
(/ (/ (/ 2 t) (sin k)) (sqrt (fma (/ k t) (/ k t) 2)))
9.6b
(/ (/ (* (/ l t) (/ l t)) (tan k)) (sqrt (fma (/ k t) (/ k t) 2)))
7.1b
(sqrt (fma (/ k t) (/ k t) 2))
7.1b
(sqrt (fma (/ k t) (/ k t) 2))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 244
Calls

4 calls. Slowest were:

19.0ms
(/ (/ (* (/ l t) (/ l t)) (tan k)) (sqrt (fma (/ k t) (/ k t) 2)))
14.0ms
(/ (/ (/ 2 t) (sin k)) (sqrt (fma (/ k t) (/ k t) 2)))
2.0ms
(sqrt (fma (/ k t) (/ k t) 2))

series412.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

205.0ms
(/ (/ (* (/ l t) (/ l t)) (tan k)) (sqrt (fma (/ k t) (/ k t) 2)))
130.0ms
(/ (/ (/ 2 t) (sin k)) (sqrt (fma (/ k t) (/ k t) 2)))
40.0ms
(sqrt (fma (/ k t) (/ k t) 2))
37.0ms
(sqrt (fma (/ k t) (/ k t) 2))

simplify19.5s

Counts
354 → 256
Calls

354 calls. Slowest were:

441.0ms
(/ (/ (/ l t) (cbrt (tan k))) (sqrt (fma (/ k t) (/ k t) 2)))
360.0ms
(/ (/ (* (/ l t) (/ l t)) (sin k)) (sqrt (* (cbrt (fma (/ k t) (/ k t) 2)) (cbrt (fma (/ k t) (/ k t) 2)))))
350.0ms
(/ (/ (* (/ l t) (/ l t)) (tan k)) (sqrt (cbrt (fma (/ k t) (/ k t) 2))))

prune4.8s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 9.9b

localize36.0ms

Local error

Found 4 expressions with local error:

11.3b
(/ (/ (/ 2 t) (sin k)) (sqrt (fma (/ k t) (/ k t) 2)))
10.6b
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
10.6b
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
7.1b
(sqrt (fma (/ k t) (/ k t) 2))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 185
Calls

4 calls. Slowest were:

14.0ms
(/ (/ (/ 2 t) (sin k)) (sqrt (fma (/ k t) (/ k t) 2)))
2.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
2.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))

series481.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

161.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
143.0ms
(/ (/ (/ 2 t) (sin k)) (sqrt (fma (/ k t) (/ k t) 2)))
138.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
39.0ms
(sqrt (fma (/ k t) (/ k t) 2))

simplify11.0s

Counts
250 → 197
Calls

250 calls. Slowest were:

528.0ms
(exp (* 1/4 (- (* 2 (log (/ -1 t))) (* 2 (log (/ -1 k))))))
453.0ms
(exp (* 1/4 (- (* 2 (log (/ -1 t))) (* 2 (log (/ -1 k))))))
312.0ms
(/ (/ (* (cbrt (/ 2 t)) (cbrt (/ 2 t))) (* (cbrt (sin k)) (cbrt (sin k)))) (* (cbrt (sqrt (fma (/ k t) (/ k t) 2))) (cbrt (sqrt (fma (/ k t) (/ k t) 2)))))

prune4.2s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 9.9b

localize24.0ms

Local error

Found 4 expressions with local error:

10.6b
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
10.6b
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
10.6b
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
10.6b
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

1.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
1.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
1.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))

series591.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

160.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
157.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
139.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
135.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))

simplify3.5s

Counts
44 → 84
Calls

44 calls. Slowest were:

498.0ms
(exp (* 1/4 (- (* 2 (log (/ -1 t))) (* 2 (log (/ -1 k))))))
468.0ms
(exp (* 1/4 (- (* 2 (log (/ -1 t))) (* 2 (log (/ -1 k))))))
454.0ms
(exp (* 1/4 (- (* 2 (log (/ -1 t))) (* 2 (log (/ -1 k))))))

prune2.0s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 9.9b

regimes479.0ms

Accuracy

0% (1.6b remaining)

Error of 12.7b against oracle of 11.1b and baseline of 12.7b

bsearch5.0ms