Average Error: 42.3 → 9.5
Time: 47.4s
Precision: 64
Internal Precision: 128
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -9.370952148789538 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -2.560939417405444 \cdot 10^{-203}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}}\\ \mathbf{elif}\;t \le -7.350478852194393 \cdot 10^{-299}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 5.5676703725785085 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}}\\ \mathbf{elif}\;t \le 1.1716671711905764 \cdot 10^{-156} \lor \neg \left(t \le 1.2659068094957783 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right) + \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\ell \cdot \left(\frac{1}{x} \cdot \left(\ell \cdot 2\right)\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\\ \end{array}\]

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 5 regimes

  2. if t < -9.370952148789538e+89 or -2.560939417405444e-203 < t < -7.350478852194393e-299

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 10.3

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

    3. Simplified10.3

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

    if -9.370952148789538e+89 < t < -2.560939417405444e-203

    1. Initial program 29.7

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

    2. Taylor expanded around inf 13.8

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

    3. Simplified8.5

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

    5. Applied add-cube-cbrt8.5

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

    6. Applied associate-*l*8.5

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

    if -7.350478852194393e-299 < t < 5.5676703725785085e-183

    1. Initial program 61.2

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

    2. Taylor expanded around inf 32.7

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

    3. Simplified30.7

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

    5. Applied add-sqr-sqrt30.7

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

    6. Applied associate-*l*30.7

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

    if 5.5676703725785085e-183 < t < 1.1716671711905764e-156 or 1.2659068094957783e+137 < t

    1. Initial program 57.9

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

    2. Taylor expanded around inf 4.9

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

    3. Simplified4.9

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

    if 1.1716671711905764e-156 < t < 1.2659068094957783e+137

    1. Initial program 24.5

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

    2. Taylor expanded around inf 10.8

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

    3. Simplified5.4

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

    5. Applied div-inv5.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\ell \cdot \frac{1}{x}\right)} \cdot \left(\ell \cdot 2\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]

    6. Applied associate-*l*5.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\ell \cdot \left(\frac{1}{x} \cdot \left(\ell \cdot 2\right)\right)} + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.370952148789538 \cdot 10^{+89}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -2.560939417405444 \cdot 10^{-203}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}}\\ \mathbf{elif}\;t \le -7.350478852194393 \cdot 10^{-299}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 5.5676703725785085 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right) + \frac{\ell}{x} \cdot \left(\ell \cdot 2\right)}}\\ \mathbf{elif}\;t \le 1.1716671711905764 \cdot 10^{-156} \lor \neg \left(t \le 1.2659068094957783 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}} \cdot \left(t - \frac{t}{2}\right) + \left(\sqrt{2} + \frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\ell \cdot \left(\frac{1}{x} \cdot \left(\ell \cdot 2\right)\right) + \left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(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: 33.0s)Debug log

start511.0ms

Algorithm
intervals

setup331.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 43.4b

localize75.0ms

Local error

Found 4 expressions with local error:

19.9b
(sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))
14.6b
(- (* (/ (+ 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))))

rewrite62.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

21.0ms
(sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))
14.0ms
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))
14.0ms
(- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))

series1.4s

Counts
4 → 12
Calls

4 calls. Slowest were:

946.0ms
(sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))
250.0ms
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))
167.0ms
(- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))
35.0ms
(* (sqrt 2) t)

simplify8.1s

Counts
46 → 78
Calls

46 calls. Slowest were:

506.0ms
(- (+ (* 2 (pow t 2)) (+ (* 2 (* x (pow l 2))) (* 2 (pow l 2)))))
462.0ms
(- (* (sqrt -2) t) (+ (* 2 (/ (* t x) (sqrt -2))) (+ (* 2 (/ (* t (pow x 2)) (sqrt -2))) (* 2 (/ (* t (pow x 2)) (pow (sqrt -2) 3))))))
335.0ms
(/ (sqrt 2) (sqrt (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))

prune1.4s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 3.6b

localize23.0ms

Local error

Found 4 expressions with local error:

25.5b
(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))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

6.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
5.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
3.0ms
(* (/ l x) (* l 2))

series697.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

604.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
41.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
40.0ms
(* (sqrt 2) t)
12.0ms
(* (/ l x) (* l 2))

simplify3.4s

Counts
33 → 69
Calls

33 calls. Slowest were:

632.0ms
(* (* (* (/ l x) (/ l x)) (/ l x)) (* (* (* l 2) (* l 2)) (* l 2)))
512.0ms
(sqrt (+ (* (* l (* l 2)) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t)))))
482.0ms
(* x (+ (* 2 2) (- (* (/ 4 x) (/ 4 x)) (* 2 (/ 4 x)))))

prune986.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 3.1b

localize14.0ms

Local error

Found 4 expressions with local error:

25.5b
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
0.3b
(* (* (cbrt (sqrt 2)) (cbrt (sqrt 2))) (* (cbrt (sqrt 2)) t))
0.2b
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
0.2b
(* (cbrt (sqrt 2)) t)

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

15.0ms
(* (* (cbrt (sqrt 2)) (cbrt (sqrt 2))) (* (cbrt (sqrt 2)) t))
6.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
5.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))

series892.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

524.0ms
(* (cbrt (sqrt 2)) t)
263.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
63.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
41.0ms
(* (* (cbrt (sqrt 2)) (cbrt (sqrt 2))) (* (cbrt (sqrt 2)) t))

simplify3.8s

Counts
37 → 71
Calls

37 calls. Slowest were:

475.0ms
(+ (* (* l (* l 2)) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t))))
467.0ms
(* (* (* (* (cbrt (sqrt 2)) (cbrt (sqrt 2))) (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (* (* (cbrt (sqrt 2)) t) (* (cbrt (sqrt 2)) t)) (* (cbrt (sqrt 2)) t)))
444.0ms
(* x (+ (* 2 2) (- (* (/ 4 x) (/ 4 x)) (* 2 (/ 4 x)))))

prune1.4s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 3.1b

localize16.0ms

Local error

Found 4 expressions with local error:

25.5b
(sqrt (+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t))))
0.4b
(* (sqrt 2) t)
0.4b
(* (/ 1 x) (* l 2))
0.2b
(+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t)))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

10.0ms
(+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t)))
4.0ms
(sqrt (+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t))))
3.0ms
(* (/ 1 x) (* l 2))

series349.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

226.0ms
(sqrt (+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t))))
65.0ms
(+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t)))
40.0ms
(* (sqrt 2) t)
18.0ms
(* (/ 1 x) (* l 2))

simplify4.5s

Counts
33 → 69
Calls

33 calls. Slowest were:

673.0ms
(* (* (* (/ 1 x) (/ 1 x)) (/ 1 x)) (* (* (* l 2) (* l 2)) (* l 2)))
551.0ms
(sqrt (+ (* (* l (* 1 (* l 2))) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t)))))
524.0ms
(+ (* (* l (* 1 (* l 2))) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t))))

prune1.2s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 3.1b

regimes718.0ms

Accuracy

74% (6.4b remaining)

Error of 9.5b against oracle of 3.1b and baseline of 27.6b

bsearch3.1s