Average Error: 42.0 → 9.3
Time: 41.3s
Precision: 64
Internal Precision: 128
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.07394485735048 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -3.563799924669161 \cdot 10^{-160}:\\ \;\;\;\;\frac{\left(\left(\sqrt{\sqrt{2}} \cdot t\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\right) \cdot \left|\sqrt[3]{\sqrt{2}}\right|}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{elif}\;t \le -1.8401747941686373 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 2.590623709549756 \cdot 10^{+63}:\\ \;\;\;\;\frac{t \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right) + \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if t < -5.07394485735048e+81 or -3.563799924669161e-160 < t < -1.8401747941686373e-280

    1. Initial program 51.1

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around -inf 11.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}} + \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)\right)}}\]

    3. Simplified11.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{2}{\sqrt{2}} \cdot \left(\frac{\frac{t}{x}}{x \cdot 2} - \frac{t}{x}\right) - \left(\frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}} + \sqrt{2} \cdot t\right)}}\]

    if -5.07394485735048e+81 < t < -3.563799924669161e-160

    1. Initial program 26.0

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 9.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified4.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt4.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot t}{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]

    6. Applied associate-*l*4.1

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}}{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt4.1

      \[\leadsto \frac{\sqrt{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]

    9. Applied sqrt-prod4.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\right)} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]

    10. Applied associate-*l*4.1

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \left(\sqrt{\sqrt[3]{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)\right)}}{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]

    11. Simplified4.1

      \[\leadsto \frac{\color{blue}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \left(\sqrt{\sqrt[3]{\sqrt{2}}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)\right)}{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]

    if -1.8401747941686373e-280 < t < 2.590623709549756e+63

    1. Initial program 40.6

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 19.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified14.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt14.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot t}{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]

    6. Applied associate-*l*14.8

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}}{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]
    7. Using strategy rm

    8. Applied associate-*r*14.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right) \cdot t}}{\sqrt{\frac{\ell}{x} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]

    if 2.590623709549756e+63 < t

    1. Initial program 44.9

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 4.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}} + \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]

    3. Simplified4.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right) + t \cdot \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.07394485735048 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -3.563799924669161 \cdot 10^{-160}:\\ \;\;\;\;\frac{\left(\left(\sqrt{\sqrt{2}} \cdot t\right) \cdot \sqrt{\sqrt[3]{\sqrt{2}}}\right) \cdot \left|\sqrt[3]{\sqrt{2}}\right|}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{elif}\;t \le -1.8401747941686373 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 2.590623709549756 \cdot 10^{+63}:\\ \;\;\;\;\frac{t \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{\left(2 \cdot \ell\right) \cdot \frac{\ell}{x} + \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right) + \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))

Details

Time bar (total: 40.0s)Debug log

sample428.0ms

Algorithm
intervals

simplify316.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

315.0ms
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 44.7b

localize77.0ms

Local error

Found 4 expressions with local error:

22.3b
(sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))
10.5b
(- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))
0.4b
(* (sqrt 2) t)
0.0b
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))

rewrite91.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

36.0ms
(sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))
30.0ms
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))
17.0ms
(- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))

series1.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

921.0ms
(sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))
276.0ms
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))
121.0ms
(- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))
23.0ms
(* (sqrt 2) t)

simplify8.2s

Counts
46 → 78
Calls

46 calls. Slowest were:

456.0ms
(- (+ (* 2 (pow t 2)) (+ (* 2 (* x (pow l 2))) (* 2 (pow l 2)))))
378.0ms
(- (* (sqrt -2) t) (+ (* 2 (/ (* t x) (sqrt -2))) (+ (* 2 (/ (* t (pow x 2)) (sqrt -2))) (* 2 (/ (* t (pow x 2)) (pow (sqrt -2) 3))))))
341.0ms
(- (log (* (sqrt 2) t)) (log (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))

prune1.1s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 3.8b

localize47.0ms

Local error

Found 4 expressions with local error:

30.5b
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
0.4b
(* (sqrt 2) t)
0.2b
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
0.1b
(* (/ l x) (* l 2))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

14.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
11.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
7.0ms
(* (/ l x) (* l 2))

series331.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

256.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
38.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
23.0ms
(* (sqrt 2) t)
13.0ms
(* (/ l x) (* l 2))

simplify3.7s

Counts
33 → 69
Calls

33 calls. Slowest were:

838.0ms
(+ (* (* l (* l 2)) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t))))
581.0ms
(* (* (* (/ l x) (/ l x)) (/ l x)) (* (* (* l 2) (* l 2)) (* l 2)))
388.0ms
(sqrt (+ (* (* l (* l 2)) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t)))))

prune1.0s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 2.9b

localize25.0ms

Local error

Found 4 expressions with local error:

30.5b
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
0.3b
(* (sqrt (sqrt 2)) (* (sqrt (sqrt 2)) t))
0.2b
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
0.2b
(* (sqrt (sqrt 2)) t)

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

12.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
9.0ms
(* (sqrt (sqrt 2)) (* (sqrt (sqrt 2)) t))
7.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))

series413.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

263.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
67.0ms
(* (sqrt (sqrt 2)) t)
44.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
39.0ms
(* (sqrt (sqrt 2)) (* (sqrt (sqrt 2)) t))

simplify2.9s

Counts
38 → 73
Calls

38 calls. Slowest were:

397.0ms
(* x (+ (* 2 2) (- (* (/ 4 x) (/ 4 x)) (* 2 (/ 4 x)))))
395.0ms
(sqrt (+ (* (* l (* l 2)) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t)))))
377.0ms
(+ (* (* l (* l 2)) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t))))

prune1.1s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 2.9b

localize31.0ms

Local error

Found 4 expressions with local error:

30.5b
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
0.3b
(* (fabs (cbrt (sqrt 2))) (* (sqrt (cbrt (sqrt 2))) (* (sqrt (sqrt 2)) t)))
0.3b
(* (sqrt (cbrt (sqrt 2))) (* (sqrt (sqrt 2)) t))
0.2b
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))

rewrite77.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

38.0ms
(* (fabs (cbrt (sqrt 2))) (* (sqrt (cbrt (sqrt 2))) (* (sqrt (sqrt 2)) t)))
18.0ms
(* (sqrt (cbrt (sqrt 2))) (* (sqrt (sqrt 2)) t))
12.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))

series1.5s

Counts
4 → 12
Calls

4 calls. Slowest were:

622.0ms
(* (fabs (cbrt (sqrt 2))) (* (sqrt (cbrt (sqrt 2))) (* (sqrt (sqrt 2)) t)))
556.0ms
(* (sqrt (cbrt (sqrt 2))) (* (sqrt (sqrt 2)) t))
291.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
58.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))

simplify3.3s

Counts
45 → 80
Calls

45 calls. Slowest were:

429.0ms
(sqrt (+ (* (* l (* l 2)) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t)))))
406.0ms
(+ (* (* l (* l 2)) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t))))
372.0ms
(* x (+ (* 2 2) (- (* (/ 4 x) (/ 4 x)) (* 2 (/ 4 x)))))

prune1.6s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 2.9b

regimes944.0ms

Accuracy

73.9% (6.2b remaining)

Error of 9.3b against oracle of 3.1b and baseline of 26.9b

bsearch2.9s

end0.0ms

sample8.5s

Algorithm
intervals