Average Error: 32.4 → 12.5
Time: 2.1m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{\ell}{t}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right) \cdot \frac{\frac{2}{t}}{\frac{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}{\frac{1}{\sin k}}}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 32.4

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

  2. Initial simplification24.7

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

  4. Applied add-sqr-sqrt24.7

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

  5. Applied times-frac18.8

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

  6. Applied times-frac16.7

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

  8. Applied add-sqr-sqrt16.6

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

  9. Applied *-un-lft-identity16.6

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

  10. Applied times-frac13.7

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

  11. Applied times-frac12.6

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

  12. Simplified12.6

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

  14. Applied div-inv12.6

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

  15. Applied associate-/l*12.5

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

  16. Final simplification12.5

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

Reproduce

herbie shell --seed 2018365 +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: 1.7m)Debug log

start756.0ms

Algorithm
intervals

setup225.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 22.7b

localize38.0ms

Local error

Found 4 expressions with local error:

15.1b
(/ (/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k))) (fma (/ k t) (/ k t) 2))
10.2b
(/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k)))
0.6b
(* (/ 2 t) (* (/ l t) (/ l t)))
0.2b
(* (sin k) (tan k))

rewrite56.0ms

Algorithm
rewrite-expression-head
Counts
4 → 137
Calls

4 calls. Slowest were:

23.0ms
(* (/ 2 t) (* (/ l t) (/ l t)))
14.0ms
(/ (/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k))) (fma (/ k t) (/ k t) 2))
14.0ms
(/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k)))

series555.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

266.0ms
(/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k)))
209.0ms
(/ (/ (* (/ 2 t) (* (/ l t) (/ l t))) (* (sin k) (tan k))) (fma (/ k t) (/ k t) 2))
57.0ms
(* (sin k) (tan k))
22.0ms
(* (/ 2 t) (* (/ l t) (/ l t)))

simplify23.8s

Counts
121 → 149
Calls

121 calls. Slowest were:

952.0ms
(/ (/ (* (* (* (/ 2 t) (* (/ l t) (/ l t))) (* (/ 2 t) (* (/ l t) (/ l t)))) (* (/ 2 t) (* (/ l t) (/ l t)))) (* (* (* (sin k) (tan k)) (* (sin k) (tan k))) (* (sin k) (tan k)))) (* (* (fma (/ k t) (/ k t) 2) (fma (/ k t) (/ k t) 2)) (fma (/ k t) (/ k t) 2)))
831.0ms
(/ (/ (* (* (* (/ 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)))
730.0ms
(/ (/ (* (* (* (/ 2 t) (/ 2 t)) (/ 2 t)) (* (* (* (/ l t) (/ l t)) (/ l t)) (* (* (/ l t) (/ l t)) (/ l t)))) (* (* (* (sin k) (tan k)) (* (sin k) (tan k))) (* (sin k) (tan k)))) (* (* (fma (/ k t) (/ k t) 2) (fma (/ k t) (/ k t) 2)) (fma (/ k t) (/ k t) 2)))

prune2.5s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 11.4b

localize23.0ms

Local error

Found 4 expressions with local error:

11.6b
(/ (/ (/ 2 t) (sin k)) (sqrt (fma (/ k t) (/ k t) 2)))
11.1b
(/ (/ (* (/ l t) (/ l t)) (tan k)) (sqrt (fma (/ k t) (/ k t) 2)))
8.0b
(sqrt (fma (/ k t) (/ k t) 2))
8.0b
(sqrt (fma (/ k t) (/ k t) 2))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 244
Calls

4 calls. Slowest were:

10.0ms
(/ (/ (* (/ l t) (/ l t)) (tan k)) (sqrt (fma (/ k t) (/ k t) 2)))
6.0ms
(/ (/ (/ 2 t) (sin k)) (sqrt (fma (/ k t) (/ k t) 2)))
2.0ms
(sqrt (fma (/ k t) (/ k t) 2))

series331.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

165.0ms
(/ (/ (* (/ l t) (/ l t)) (tan k)) (sqrt (fma (/ k t) (/ k t) 2)))
119.0ms
(/ (/ (/ 2 t) (sin k)) (sqrt (fma (/ k t) (/ k t) 2)))
24.0ms
(sqrt (fma (/ k t) (/ k t) 2))
23.0ms
(sqrt (fma (/ k t) (/ k t) 2))

simplify19.3s

Counts
354 → 256
Calls

354 calls. Slowest were:

562.0ms
(- (log (/ (* (/ l t) (/ l t)) (tan k))) (log (sqrt (fma (/ k t) (/ k t) 2))))
367.0ms
(/ (/ (* (/ l t) (/ l t)) (sin k)) (sqrt (* (cbrt (fma (/ k t) (/ k t) 2)) (cbrt (fma (/ k t) (/ k t) 2)))))
358.0ms
(/ (/ (* (/ l t) (/ l t)) (tan k)) (sqrt (sqrt (fma (/ k t) (/ k t) 2))))

prune4.8s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 10.7b

localize25.0ms

Local error

Found 4 expressions with local error:

11.6b
(/ (/ (/ 2 t) (sin k)) (sqrt (fma (/ k t) (/ k t) 2)))
11.1b
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
11.1b
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
8.0b
(sqrt (fma (/ k t) (/ k t) 2))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 185
Calls

4 calls. Slowest were:

6.0ms
(/ (/ (/ 2 t) (sin k)) (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)))

series478.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

150.0ms
(/ (/ (/ 2 t) (sin k)) (sqrt (fma (/ k t) (/ k t) 2)))
149.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
144.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
35.0ms
(sqrt (fma (/ k t) (/ k t) 2))

simplify10.6s

Counts
250 → 197
Calls

250 calls. Slowest were:

469.0ms
(exp (* 1/4 (- (* 2 (log (/ -1 t))) (* 2 (log (/ -1 k))))))
439.0ms
(exp (* 1/4 (- (* 2 (log (/ -1 t))) (* 2 (log (/ -1 k))))))
282.0ms
(/ (/ (* (cbrt (/ 2 t)) (cbrt (/ 2 t))) 1) (* (cbrt (sqrt (fma (/ k t) (/ k t) 2))) (cbrt (sqrt (fma (/ k t) (/ k t) 2)))))

prune4.0s

Pruning

14 alts after pruning (13 fresh and 1 done)

Merged error: 10.7b

localize17.0ms

Local error

Found 4 expressions with local error:

11.1b
(/ (/ 2 t) (/ (sqrt (fma (/ k t) (/ k t) 2)) (/ 1 (sin k))))
11.1b
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
11.1b
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
8.0b
(sqrt (fma (/ k t) (/ k t) 2))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 218
Calls

4 calls. Slowest were:

11.0ms
(/ (/ 2 t) (/ (sqrt (fma (/ k t) (/ k t) 2)) (/ 1 (sin k))))
7.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
2.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))

series471.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

167.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
154.0ms
(sqrt (sqrt (fma (/ k t) (/ k t) 2)))
116.0ms
(/ (/ 2 t) (/ (sqrt (fma (/ k t) (/ k t) 2)) (/ 1 (sin k))))
34.0ms
(sqrt (fma (/ k t) (/ k t) 2))

simplify26.4s

Counts
301 → 230
Calls

301 calls. Slowest were:

783.0ms
(/ (/ 2 t) (/ (sqrt (fma (/ k t) (/ k t) 2)) (/ 1 (sin k))))
510.0ms
(/ (* (cbrt (/ 2 t)) (cbrt (/ 2 t))) (/ (* (cbrt (sqrt (fma (/ k t) (/ k t) 2))) (cbrt (sqrt (fma (/ k t) (/ k t) 2)))) (* (cbrt (/ 1 (sin k))) (cbrt (/ 1 (sin k))))))
506.0ms
(/ (/ 2 t) (/ (sqrt (fma (/ k t) (/ k t) 2)) (/ 1 (sin k))))

prune4.6s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 10.7b

regimes373.0ms

Accuracy

0% (1.6b remaining)

Error of 12.5b against oracle of 10.9b and baseline of 12.5b

bsearch2.0ms