Average Error: 47.2 → 18.5
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 0.0:\\ \;\;\;\;\frac{2}{(\left(\frac{\frac{t}{\ell}}{\ell}\right) \cdot \left({k}^{6} \cdot \frac{1}{6}\right) + \left(\left({k}^{4} + \frac{31}{360} \cdot {k}^{8}\right) \cdot \frac{\frac{t}{\ell}}{\ell}\right))_*}\\ \mathbf{elif}\;\ell \cdot \ell \le 2.724912876006757 \cdot 10^{+292}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{{\ell}^{2}} \cdot \frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) - 1\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if (* l l) < 0.0

    1. Initial program 45.4

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

    2. Taylor expanded around -inf 20.4

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

    4. Applied associate-*r*19.6

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

    5. Taylor expanded around 0 22.3

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{6} \cdot \frac{t \cdot {k}^{6}}{{\ell}^{2}} + \left(\frac{31}{360} \cdot \frac{t \cdot {k}^{8}}{{\ell}^{2}} + \frac{t \cdot {k}^{4}}{{\ell}^{2}}\right)}}\]

    6. Simplified19.1

      \[\leadsto \frac{2}{\color{blue}{(\left(\frac{\frac{t}{\ell}}{\ell}\right) \cdot \left({k}^{6} \cdot \frac{1}{6}\right) + \left(\frac{\frac{t}{\ell}}{\ell} \cdot \left(\frac{31}{360} \cdot {k}^{8} + {k}^{4}\right)\right))_*}}\]

    if 0.0 < (* l l) < 2.724912876006757e+292

    1. Initial program 43.7

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

    2. Taylor expanded around -inf 15.2

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

    4. Applied associate-*r*13.0

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

    6. Applied unpow213.0

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

    7. Applied associate-*r*9.2

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

    9. Applied times-frac7.5

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

    if 2.724912876006757e+292 < (* l l)

    1. Initial program 61.7

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

    3. Applied cube-mult61.7

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

    4. Applied times-frac51.7

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

    5. Applied associate-*l*51.7

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

  4. Final simplification18.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 0.0:\\ \;\;\;\;\frac{2}{(\left(\frac{\frac{t}{\ell}}{\ell}\right) \cdot \left({k}^{6} \cdot \frac{1}{6}\right) + \left(\left({k}^{4} + \frac{31}{360} \cdot {k}^{8}\right) \cdot \frac{\frac{t}{\ell}}{\ell}\right))_*}\\ \mathbf{elif}\;\ell \cdot \ell \le 2.724912876006757 \cdot 10^{+292}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{{\ell}^{2}} \cdot \frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) - 1\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 +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: 1.5m)Debug log

sample1.1s

Algorithm
intervals
Results
434×(pre true 80)
140×(body real 1280)
76×(body real 640)
70×(body real 2560)
68×(body real 80)
34×(body real 5120)
28×(body real 320)
18×(body real 160)

simplify84.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
84.0ms
(/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1)))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 44.1b

localize59.0ms

Local error

Found 4 expressions with local error:

30.3b
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
16.2b
(/ (pow t 3) (* l l))
1.9b
(- (+ 1 (pow (/ k t) 2)) 1)
1.0b
(* (/ (pow t 3) (* l l)) (sin k))

rewrite97.0ms

Algorithm
rewrite-expression-head
Rules
33×add-exp-log
25×add-cbrt-cube
15×prod-exp
15×pow1
13×associate-*l/
13×add-sqr-sqrt
11×cbrt-unprod
11×add-cube-cbrt
11×*-un-lft-identity
10×associate-*l*
10×times-frac
9×frac-times
7×pow-prod-down
7×associate-*r*
6×tan-quot
6×div-exp
5×associate-/l*
5×flip--
5×associate-*r/
5×flip3--
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate--l+
3×unpow3
3×fma-neg
3×distribute-lft-in
3×sub-neg
3×cbrt-undiv
3×pow-to-exp
3×cube-mult
3×distribute-rgt-in
2×div-inv
2×*-commutative
2×difference-of-sqr-1
1×frac-2neg
1×associate-/r*
1×clear-num
1×expm1-def
Counts
4 → 118
Calls
4 calls:
Slowest
72.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
13.0ms
(- (+ 1 (pow (/ k t) 2)) 1)
5.0ms
(* (/ (pow t 3) (* l l)) (sin k))
5.0ms
(/ (pow t 3) (* l l))

series418.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
305.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
75.0ms
(* (/ (pow t 3) (* l l)) (sin k))
21.0ms
(- (+ 1 (pow (/ k t) 2)) 1)
17.0ms
(/ (pow t 3) (* l l))

simplify18.9s

Counts
90 → 130
Calls
90 calls:
Slowest
1.3s
(* (* (* (pow t 3) (sin k)) (sin k)) (- (+ 1 (pow (/ k t) 2)) 1))
1.1s
(cbrt (- (+ 1 (pow (/ k t) 2)) 1))
884.0ms
(sqrt (- (+ 1 (pow (/ k t) 2)) 1))
797.0ms
(- (+ 1 (pow (/ k t) 2)) 1)
751.0ms
(* (* l l) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (+ (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))

prune2.2s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 16.7b

localize27.0ms

Local error

Found 4 expressions with local error:

20.5b
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
4.6b
(* t (* (pow k 2) (pow (sin k) 2)))
0.5b
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
0.4b
(pow (sin k) 2)

rewrite36.0ms

Algorithm
rewrite-expression-head
Rules
12×add-exp-log
9×add-cbrt-cube
6×add-cube-cbrt
6×*-un-lft-identity
6×associate-/r*
6×add-sqr-sqrt
5×pow1
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×div-inv
3×div-exp
3×unpow-prod-down
2×prod-exp
2×times-frac
2×frac-2neg
2×clear-num
2×cbrt-undiv
2×pow-to-exp
1×unpow2
1×associate-/l*
1×cbrt-unprod
1×*-commutative
1×associate-/r/
1×pow-exp
1×associate-*r*
1×pow-pow
Counts
4 → 65
Calls
4 calls:
Slowest
15.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
15.0ms
(* t (* (pow k 2) (pow (sin k) 2)))
3.0ms
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
2.0ms
(pow (sin k) 2)

series652.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
281.0ms
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
275.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
82.0ms
(* t (* (pow k 2) (pow (sin k) 2)))
15.0ms
(pow (sin k) 2)

simplify5.5s

Counts
32 → 77
Calls
32 calls:
Slowest
597.0ms
(cbrt (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
579.0ms
(sqrt (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
517.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
514.0ms
(/ (* (* (* t (* (pow k 2) (pow (sin k) 2))) (* t (* (pow k 2) (pow (sin k) 2)))) (* t (* (pow k 2) (pow (sin k) 2)))) (* (* (* (pow l 2) (pow l 2)) (pow l 2)) (* (* (cos k) (cos k)) (cos k))))
511.0ms
(/ (* (* (* t (* (pow k 2) (pow (sin k) 2))) (* t (* (pow k 2) (pow (sin k) 2)))) (* t (* (pow k 2) (pow (sin k) 2)))) (* (* (* (pow l 2) (cos k)) (* (pow l 2) (cos k))) (* (pow l 2) (cos k))))

prune1.4s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 14.4b

localize18.0ms

Local error

Found 4 expressions with local error:

20.5b
(/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k)))
5.5b
(* t (pow k 2))
0.5b
(/ 2 (/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k))))
0.4b
(pow (sin k) 2)

rewrite29.0ms

Algorithm
rewrite-expression-head
Rules
39×add-exp-log
17×prod-exp
16×add-cbrt-cube
12×div-exp
8×pow-to-exp
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×associate-/r*
5×pow1
4×add-log-exp
4×log1p-expm1-u
4×cbrt-unprod
4×pow-exp
4×associate-*r*
4×cbrt-undiv
4×expm1-log1p-u
3×div-inv
3×unpow-prod-down
2×unpow2
2×times-frac
2×frac-2neg
2×clear-num
1×associate-/l*
1×*-commutative
1×associate-/r/
1×pow-pow
Counts
4 → 79
Calls
4 calls:
Slowest
21.0ms
(/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k)))
3.0ms
(/ 2 (/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k))))
2.0ms
(* t (pow k 2))
2.0ms
(pow (sin k) 2)

series521.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
257.0ms
(/ 2 (/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k))))
220.0ms
(/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k)))
24.0ms
(pow (sin k) 2)
20.0ms
(* t (pow k 2))

simplify5.8s

Counts
46 → 91
Calls
46 calls:
Slowest
571.0ms
(/ (* (* (* (* t (pow k 2)) (* t (pow k 2))) (* t (pow k 2))) (* (* (pow (sin k) 2) (pow (sin k) 2)) (pow (sin k) 2))) (* (* (* (pow l 2) (pow l 2)) (pow l 2)) (* (* (cos k) (cos k)) (cos k))))
561.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
518.0ms
(/ (* (* (* (* t (pow k 2)) (pow (sin k) 2)) (* (* t (pow k 2)) (pow (sin k) 2))) (* (* t (pow k 2)) (pow (sin k) 2))) (* (* (* (pow l 2) (pow l 2)) (pow l 2)) (* (* (cos k) (cos k)) (cos k))))
483.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
481.0ms
(/ (* (* (* (* t (pow k 2)) (* t (pow k 2))) (* t (pow k 2))) (* (* (pow (sin k) 2) (pow (sin k) 2)) (pow (sin k) 2))) (* (* (* (pow l 2) (cos k)) (* (pow l 2) (cos k))) (* (pow l 2) (cos k))))

prune1.4s

Pruning

17 alts after pruning (16 fresh and 1 done)

Merged error: 12.5b

localize28.0ms

Local error

Found 4 expressions with local error:

20.5b
(/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k)))
0.5b
(/ 2 (/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k))))
0.4b
(pow (sin k) 2)
0.1b
(* (* (* t k) k) (pow (sin k) 2))

rewrite39.0ms

Algorithm
rewrite-expression-head
Rules
44×add-exp-log
20×prod-exp
18×add-cbrt-cube
12×div-exp
9×pow-to-exp
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×associate-*r*
6×unpow-prod-down
6×associate-/r*
5×cbrt-unprod
5×pow-exp
5×pow1
4×add-log-exp
4×log1p-expm1-u
4×cbrt-undiv
4×expm1-log1p-u
3×div-inv
2×unpow2
2×times-frac
2×frac-2neg
2×clear-num
1×associate-/l*
1×*-commutative
1×associate-/r/
1×associate-*l*
1×pow-pow
Counts
4 → 87
Calls
4 calls:
Slowest
27.0ms
(/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k)))
8.0ms
(* (* (* t k) k) (pow (sin k) 2))
2.0ms
(/ 2 (/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k))))
1.0ms
(pow (sin k) 2)

series622.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
283.0ms
(/ 2 (/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k))))
222.0ms
(/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k)))
90.0ms
(* (* (* t k) k) (pow (sin k) 2))
26.0ms
(pow (sin k) 2)

simplify7.1s

Counts
55 → 99
Calls
55 calls:
Slowest
603.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
599.0ms
(* (* (* (* (* t k) k) (* (* t k) k)) (* (* t k) k)) (* (* (pow (sin k) 2) (pow (sin k) 2)) (pow (sin k) 2)))
539.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
524.0ms
(/ (* (* (* (* (* t k) k) (pow (sin k) 2)) (* (* (* t k) k) (pow (sin k) 2))) (* (* (* t k) k) (pow (sin k) 2))) (* (* (* (pow l 2) (pow l 2)) (pow l 2)) (* (* (cos k) (cos k)) (cos k))))
499.0ms
(/ (* (* (* (* (* t k) k) (pow (sin k) 2)) (* (* (* t k) k) (pow (sin k) 2))) (* (* (* t k) k) (pow (sin k) 2))) (* (* (* (pow l 2) (cos k)) (* (pow l 2) (cos k))) (* (pow l 2) (cos k))))

prune1.5s

Pruning

20 alts after pruning (19 fresh and 1 done)

Merged error: 12.2b

regimes845.0ms

Accuracy

17.8% (6.6b remaining)

Error of 18.5b against oracle of 11.9b and baseline of 19.9b

bsearch15.0ms

end0.0ms

sample41.3s

Algorithm
intervals
Results
13267×(pre true 80)
4079×(body real 1280)
2526×(body real 640)
2193×(body real 2560)
2012×(body real 80)
1047×(body real 320)
934×(body real 5120)
476×(body real 160)