Average Error: 42.9 → 7.6
Time: 37.7s
Precision: 64
Internal Precision: 128
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.00212864258164 \cdot 10^{+19}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -1.302324579260043 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\ell \cdot 2\right) \cdot \frac{1}{x}\right) \cdot \ell + \left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right)}}\\ \mathbf{elif}\;t \le 4.548562403106992 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(\ell \cdot 2\right) \cdot \ell + \left(\left(x \cdot t\right) \cdot t\right) \cdot \left(\frac{4}{x} + 2\right)\right) \cdot \left(2 - \frac{4}{x}\right)}}{\sqrt{\left(2 - \frac{4}{x}\right) \cdot x}}}\\ \mathbf{elif}\;t \le 3.628770648199889 \cdot 10^{+88}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\ell \cdot 2\right) \cdot \frac{1}{x}\right) \cdot \ell + \left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) + \left(t - \frac{t}{2}\right) \cdot \frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}}}\\ \end{array}\]

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 < -5.00212864258164e+19

    1. Initial program 42.9

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

    2. Taylor expanded around -inf 4.7

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

    3. Simplified4.7

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

    if -5.00212864258164e+19 < t < -1.302324579260043e-143 or 4.548562403106992e-161 < t < 3.628770648199889e+88

    1. Initial program 27.5

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

    2. Taylor expanded around -inf 9.6

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

    3. Simplified4.8

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

    5. Applied div-inv4.8

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

    6. Applied associate-*l*4.8

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

    if -1.302324579260043e-143 < t < 4.548562403106992e-161

    1. Initial program 60.2

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

    2. Taylor expanded around -inf 31.6

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

    3. Simplified29.9

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

    5. Applied div-inv29.9

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

    6. Applied associate-*l*29.9

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

    8. Applied flip-+29.9

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

    9. Applied associate-*l/29.9

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

    10. Applied associate-*l/29.9

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

    11. Applied associate-*r/31.6

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

    12. Applied frac-add31.9

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

    13. Applied sqrt-div28.5

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

    14. Simplified20.8

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

    if 3.628770648199889e+88 < t

    1. Initial program 49.4

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

    2. Taylor expanded around inf 2.8

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

    3. Simplified2.9

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.00212864258164 \cdot 10^{+19}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{x}}{2 \cdot x} - \frac{t}{x}\right) \cdot \frac{2}{\sqrt{2}} - \left(\sqrt{2} \cdot t + \frac{\frac{t}{x}}{x} \cdot \frac{2}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \le -1.302324579260043 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\ell \cdot 2\right) \cdot \frac{1}{x}\right) \cdot \ell + \left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right)}}\\ \mathbf{elif}\;t \le 4.548562403106992 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\sqrt{\left(\left(\ell \cdot 2\right) \cdot \ell + \left(\left(x \cdot t\right) \cdot t\right) \cdot \left(\frac{4}{x} + 2\right)\right) \cdot \left(2 - \frac{4}{x}\right)}}{\sqrt{\left(2 - \frac{4}{x}\right) \cdot x}}}\\ \mathbf{elif}\;t \le 3.628770648199889 \cdot 10^{+88}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\ell \cdot 2\right) \cdot \frac{1}{x}\right) \cdot \ell + \left(t \cdot t\right) \cdot \left(\frac{4}{x} + 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \left(\frac{\frac{2}{x}}{\sqrt{2}} + \sqrt{2}\right) + \left(t - \frac{t}{2}\right) \cdot \frac{\frac{\frac{2}{x}}{x}}{\sqrt{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 
(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: 36.6s)Debug log

sample349.0ms

Algorithm
intervals
Results
83.0ms287×body80nan
36.0ms39×body1280valid
30.0ms163×body80valid
16.0ms23×body640valid
15.0ms18×body1280nan
14.0ms17×body640nan
10.0ms21×body320valid
4.0ms10×body160valid
2.0ms5×body160nan
1.0ms2×body320nan

simplify324.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
324.0ms
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 43.8b

localize75.0ms

Local error

Found 4 expressions with local error:

21.5b
(sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))
12.2b
(- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))
0.5b
(* (sqrt 2) t)
0.0b
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))

rewrite75.0ms

Algorithm
rewrite-expression-head
Rules
11×add-sqr-sqrt
10×add-cube-cbrt
10×*-un-lft-identity
9×sqrt-prod
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
6×times-frac
6×associate-/r*
4×sqrt-div
4×pow1
3×flip--
3×associate-*l*
3×flip3--
2×associate-/r/
2×associate--l+
1×associate-/l*
1×div-inv
1×*-commutative
1×div-exp
1×diff-log
1×pow1/2
1×distribute-lft-in
1×frac-2neg
1×sub-neg
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
1×distribute-rgt-in
Counts
4 → 66
Calls
4 calls:
Slowest
29.0ms
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))
22.0ms
(sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))
16.0ms
(- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))
4.0ms
(* (sqrt 2) t)

series1.2s

Counts
4 → 12
Calls
4 calls:
Slowest
813.0ms
(sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))
257.0ms
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))
148.0ms
(- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))
22.0ms
(* (sqrt 2) t)

simplify7.9s

Counts
46 → 78
Calls
46 calls:
Slowest
437.0ms
(- (+ (* 2 (pow t 2)) (+ (* 2 (* x (pow l 2))) (* 2 (pow l 2)))))
362.0ms
(- (* (sqrt -2) t) (+ (* 2 (/ (* t x) (sqrt -2))) (+ (* 2 (/ (* t (pow x 2)) (sqrt -2))) (* 2 (/ (* t (pow x 2)) (pow (sqrt -2) 3))))))
311.0ms
(sqrt (cbrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))
295.0ms
(- (log (* (sqrt 2) t)) (log (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))
293.0ms
(sqrt (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: 3.9b

localize22.0ms

Local error

Found 4 expressions with local error:

28.2b
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
0.5b
(* (sqrt 2) t)
0.4b
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
0.1b
(* (/ l x) (* l 2))

rewrite25.0ms

Algorithm
rewrite-expression-head
Rules
9×associate-*l/
8×add-sqr-sqrt
7×add-cube-cbrt
7×associate-*l*
7×*-un-lft-identity
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
6×pow1
4×flip-+
4×frac-add
4×sqrt-div
4×flip3-+
3×sqrt-prod
2×*-commutative
1×div-inv
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*r*
1×pow1/2
1×sum-log
1×rem-sqrt-square
1×+-commutative
Counts
4 → 57
Calls
4 calls:
Slowest
8.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
7.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
6.0ms
(* (/ l x) (* l 2))
2.0ms
(* (sqrt 2) t)

series338.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
246.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
44.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
36.0ms
(* (sqrt 2) t)
12.0ms
(* (/ l x) (* l 2))

simplify3.1s

Counts
33 → 69
Calls
33 calls:
Slowest
596.0ms
(* (* (* (/ 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)))))
379.0ms
(* x (+ (* 2 2) (- (* (/ 4 x) (/ 4 x)) (* 2 (/ 4 x)))))
374.0ms
(+ (* (* l (* l 2)) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t))))
232.0ms
(* (exp (* (/ l x) (* l 2))) (exp (* (+ 2 (/ 4 x)) (* t t))))

prune806.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 2.9b

localize14.0ms

Local error

Found 4 expressions with local error:

28.2b
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
0.4b
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
0.3b
(* (* (cbrt (sqrt 2)) (cbrt (sqrt 2))) (* (cbrt (sqrt 2)) t))
0.2b
(* (cbrt (sqrt 2)) t)

rewrite33.0ms

Algorithm
rewrite-expression-head
Rules
9×add-exp-log
9×pow1
8×associate-*l/
8×add-sqr-sqrt
7×add-cube-cbrt
7×associate-*l*
7×add-cbrt-cube
7×*-un-lft-identity
6×add-log-exp
4×flip-+
4×frac-add
4×sqrt-div
4×flip3-+
3×cbrt-unprod
3×sqrt-prod
3×prod-exp
3×pow-prod-down
3×cbrt-prod
2×*-commutative
1×associate-*r*
1×pow1/2
1×sum-log
1×rem-sqrt-square
1×+-commutative
Counts
4 → 59
Calls
4 calls:
Slowest
16.0ms
(* (* (cbrt (sqrt 2)) (cbrt (sqrt 2))) (* (cbrt (sqrt 2)) t))
6.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
6.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
4.0ms
(* (cbrt (sqrt 2)) t)

series767.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
468.0ms
(* (cbrt (sqrt 2)) t)
217.0ms
(sqrt (+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t))))
42.0ms
(+ (* (/ l x) (* l 2)) (* (+ 2 (/ 4 x)) (* t t)))
40.0ms
(* (* (cbrt (sqrt 2)) (cbrt (sqrt 2))) (* (cbrt (sqrt 2)) t))

simplify3.8s

Counts
37 → 71
Calls
37 calls:
Slowest
486.0ms
(* (* (* (* (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)))
430.0ms
(* (* (cbrt (sqrt 2)) (cbrt (sqrt 2))) (* (cbrt (sqrt 2)) t))
411.0ms
(sqrt (+ (* (* l (* l 2)) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t)))))
364.0ms
(* x (+ (* 2 2) (- (* (/ 4 x) (/ 4 x)) (* 2 (/ 4 x)))))
342.0ms
(+ (* (* 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

localize30.0ms

Local error

Found 4 expressions with local error:

28.2b
(sqrt (+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t))))
0.5b
(* (sqrt 2) t)
0.4b
(+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t)))
0.2b
(* (/ 1 x) (* l 2))

rewrite35.0ms

Algorithm
rewrite-expression-head
Rules
9×associate-*l/
8×add-sqr-sqrt
7×add-cube-cbrt
7×associate-*l*
7×*-un-lft-identity
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
6×pow1
4×flip-+
4×associate-*r/
4×frac-add
4×sqrt-div
4×flip3-+
3×sqrt-prod
2×*-commutative
1×div-inv
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*r*
1×pow1/2
1×sum-log
1×rem-sqrt-square
1×+-commutative
Counts
4 → 57
Calls
4 calls:
Slowest
13.0ms
(+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t)))
9.0ms
(sqrt (+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t))))
9.0ms
(* (/ 1 x) (* l 2))
4.0ms
(* (sqrt 2) t)

series296.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
199.0ms
(sqrt (+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t))))
60.0ms
(+ (* l (* (/ 1 x) (* l 2))) (* (+ 2 (/ 4 x)) (* t t)))
22.0ms
(* (sqrt 2) t)
14.0ms
(* (/ 1 x) (* l 2))

simplify4.2s

Counts
33 → 69
Calls
33 calls:
Slowest
616.0ms
(* (* (* (/ 1 x) (/ 1 x)) (/ 1 x)) (* (* (* l 2) (* l 2)) (* l 2)))
518.0ms
(sqrt (+ (* (* l (* 1 (* l 2))) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t)))))
466.0ms
(+ (* (* l (* 1 (* l 2))) (- 2 (/ 4 x))) (* x (* (- (* 2 2) (* (/ 4 x) (/ 4 x))) (* t t))))
352.0ms
(* x (+ (* 2 2) (- (* (/ 4 x) (/ 4 x)) (* 2 (/ 4 x)))))
311.0ms
(* (exp (* l (* (/ 1 x) (* l 2)))) (exp (* (+ 2 (/ 4 x)) (* t t))))

prune822.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 2.9b

regimes678.0ms

Accuracy

80.9% (4.7b remaining)

Error of 7.6b against oracle of 2.9b and baseline of 27.8b

bsearch875.0ms

end0.0ms

sample8.6s

Algorithm
intervals
Results
2.1s10278×body80nan
1.3s886×body1280valid
1.1s5110×body80valid
994.0ms608×body1280nan
950.0ms1063×body640valid
848.0ms620×body640nan
445.0ms630×body320valid
197.0ms333×body320nan
110.0ms311×body160valid
103.0ms199×body160nan