Average Error: 42.9 → 9.8
Time: 37.4s
Precision: 64
Internal Precision: 128
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.00212864258164 \cdot 10^{+19}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot \left(-t\right) + -2 \cdot \frac{t}{x \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \le -1.271361517563434 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \left(\left(\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right)}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{t}{x} \cdot \left(t \cdot 4\right)\right))_*}}\\ \mathbf{elif}\;t \le -7.128698999489736 \cdot 10^{-258}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot \left(-t\right) + -2 \cdot \frac{t}{x \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \le 2.4158396745379366 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(t \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{t}{x} \cdot \left(t \cdot 4\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -5.00212864258164e+19 or -1.271361517563434e-173 < t < -7.128698999489736e-258

    1. Initial program 46.1

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

    2. Simplified46.1

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

    3. Taylor expanded around -inf 10.3

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

    if -5.00212864258164e+19 < t < -1.271361517563434e-173

    1. Initial program 32.2

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

    2. Simplified32.1

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

    3. Taylor expanded around inf 12.0

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

    4. Simplified6.3

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

    6. Applied add-cube-cbrt6.3

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

    7. Applied associate-*r*6.3

      \[\leadsto \frac{\color{blue}{\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt6.3

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

    10. Applied associate-*r*6.4

      \[\leadsto \frac{\color{blue}{\left(\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt6.4

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

    13. Applied associate-*r*6.4

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

    if -7.128698999489736e-258 < t < 2.4158396745379366e+88

    1. Initial program 40.3

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

    2. Simplified40.3

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

    3. Taylor expanded around inf 18.4

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

    4. Simplified15.3

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

    6. Applied add-cube-cbrt15.3

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

    7. Applied associate-*r*15.3

      \[\leadsto \frac{\color{blue}{\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt15.3

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

    10. Applied associate-*r*15.4

      \[\leadsto \frac{\color{blue}{\left(\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}\]
    11. Using strategy rm

    12. Applied associate-*l*15.3

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

    if 2.4158396745379366e+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. Simplified49.4

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

    3. Taylor expanded around inf 3.0

      \[\leadsto \frac{t \cdot \sqrt{2}}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]

    4. Simplified3.0

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.00212864258164 \cdot 10^{+19}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot \left(-t\right) + -2 \cdot \frac{t}{x \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \le -1.271361517563434 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \left(\left(\left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right)}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{t}{x} \cdot \left(t \cdot 4\right)\right))_*}}\\ \mathbf{elif}\;t \le -7.128698999489736 \cdot 10^{-258}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot \left(-t\right) + -2 \cdot \frac{t}{x \cdot \sqrt{2}}}\\ \mathbf{elif}\;t \le 2.4158396745379366 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(t \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\frac{t}{x} \cdot \left(t \cdot 4\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(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: 36.1s)Debug log

sample304.0ms

Algorithm
intervals
Results
90.0ms287×body80nan
37.0ms18×body1280nan
29.0ms39×body1280valid
28.0ms163×body80valid
14.0ms23×body640valid
10.0ms21×body320valid
9.0ms17×body640nan
3.0ms10×body160valid
2.0ms5×body160nan
1.0ms2×body320nan

simplify470.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
470.0ms
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))

prune38.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 43.7b

localize73.0ms

Local error

Found 4 expressions with local error:

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

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
11×add-sqr-sqrt
10×add-cube-cbrt
10×*-un-lft-identity
9×sqrt-prod
6×add-exp-log
6×add-cbrt-cube
6×times-frac
6×associate-/r*
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×associate-*r*
1×associate-/l*
1×div-inv
1×fma-udef
1×*-commutative
1×div-exp
1×pow1/2
1×frac-2neg
1×clear-num
1×rem-sqrt-square
1×cbrt-undiv
Counts
4 → 64
Calls
4 calls:
Slowest
16.0ms
(sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l))))
8.0ms
(/ (* t (sqrt 2)) (sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))))
2.0ms
(* t (sqrt 2))
0.0ms
(fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))

series645.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
322.0ms
(/ (* t (sqrt 2)) (sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))))
209.0ms
(sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l))))
89.0ms
(fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))
24.0ms
(* t (sqrt 2))

simplify11.7s

Counts
39 → 76
Calls
39 calls:
Slowest
632.0ms
(sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l))))
628.0ms
(/ (* (* (* t (sqrt 2)) (* t (sqrt 2))) (* t (sqrt 2))) (* (* (sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))) (sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l))))) (sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l))))))
555.0ms
(/ t (sqrt (* (cbrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))) (cbrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))))))
551.0ms
(sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l))))
550.0ms
(/ (sqrt 2) (sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))))

prune1.0s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 4.0b

localize40.0ms

Local error

Found 4 expressions with local error:

28.2b
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
3.4b
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
0.5b
(* t (sqrt 2))
0.2b
(* (* t 4) (/ t x))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
8×add-sqr-sqrt
7×add-cube-cbrt
7×associate-*r*
7×*-un-lft-identity
6×add-exp-log
6×add-cbrt-cube
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×sqrt-prod
2×*-commutative
1×div-inv
1×fma-udef
1×cbrt-unprod
1×associate-*r/
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×pow1/2
1×rem-sqrt-square
Counts
4 → 56
Calls
4 calls:
Slowest
3.0ms
(* (* t 4) (/ t x))
1.0ms
(* t (sqrt 2))
1.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
0.0ms
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))

series358.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
261.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
49.0ms
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
27.0ms
(* t (sqrt 2))
21.0ms
(* (* t 4) (/ t x))

simplify1.3s

Counts
26 → 68
Calls
26 calls:
Slowest
478.0ms
(* (* (* (* t 4) (* t 4)) (* t 4)) (* (* (/ t x) (/ t x)) (/ t x)))
141.0ms
(+ (* 2 (pow t 2)) (+ (* 2 (/ (pow l 2) x)) (* 4 (/ (pow t 2) x))))
121.0ms
(sqrt (* (cbrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))) (cbrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))))
112.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
108.0ms
(+ (* 2 (pow t 2)) (+ (* 2 (/ (pow l 2) x)) (* 4 (/ (pow t 2) x))))

prune915.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 3.9b

localize26.0ms

Local error

Found 4 expressions with local error:

28.2b
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
3.4b
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
0.3b
(* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (cbrt (sqrt 2)))
0.2b
(* (* t 4) (/ t x))

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
10×associate-*r*
9×add-sqr-sqrt
8×add-cube-cbrt
8×add-exp-log
8×*-un-lft-identity
8×pow1
7×add-cbrt-cube
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×sqrt-prod
3×cbrt-prod
2×cbrt-unprod
2×*-commutative
2×prod-exp
2×pow-prod-down
2×associate-*l*
1×div-inv
1×fma-udef
1×associate-*r/
1×pow1/2
1×rem-sqrt-square
Counts
4 → 63
Calls
4 calls:
Slowest
13.0ms
(* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (cbrt (sqrt 2)))
3.0ms
(* (* t 4) (/ t x))
1.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
0.0ms
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))

series326.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
220.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
65.0ms
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
27.0ms
(* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (cbrt (sqrt 2)))
14.0ms
(* (* t 4) (/ t x))

simplify1.7s

Counts
32 → 75
Calls
32 calls:
Slowest
835.0ms
(* (* (* (* t 4) (* t 4)) (* t 4)) (* (* (/ t x) (/ t x)) (/ t x)))
165.0ms
(* (* (* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2))))) (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2))))) (sqrt 2))
110.0ms
(+ (* 2 (pow t 2)) (+ (* 2 (/ (pow l 2) x)) (* 4 (/ (pow t 2) x))))
96.0ms
(sqrt (* (cbrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))) (cbrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))))
92.0ms
(sqrt (sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))))

prune1.2s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 3.9b

localize13.0ms

Local error

Found 4 expressions with local error:

28.2b
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
3.4b
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
0.4b
(* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (cbrt (cbrt (sqrt 2))) (cbrt (cbrt (sqrt 2)))))
0.2b
(* (* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (cbrt (cbrt (sqrt 2))) (cbrt (cbrt (sqrt 2))))) (cbrt (cbrt (sqrt 2))))

rewrite88.0ms

Algorithm
rewrite-expression-head
Rules
18×add-exp-log
18×pow1
11×add-cbrt-cube
10×associate-*r*
9×cbrt-unprod
9×prod-exp
9×pow-prod-down
9×add-sqr-sqrt
9×cbrt-prod
8×add-cube-cbrt
8×*-un-lft-identity
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×sqrt-prod
2×*-commutative
2×associate-*l*
1×fma-udef
1×pow1/2
1×rem-sqrt-square
Counts
4 → 71
Calls
4 calls:
Slowest
57.0ms
(* (* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (cbrt (cbrt (sqrt 2))) (cbrt (cbrt (sqrt 2))))) (cbrt (cbrt (sqrt 2))))
28.0ms
(* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (cbrt (cbrt (sqrt 2))) (cbrt (cbrt (sqrt 2)))))
1.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
0.0ms
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))

series1.1s

Counts
4 → 12
Calls
4 calls:
Slowest
698.0ms
(* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (cbrt (cbrt (sqrt 2))) (cbrt (cbrt (sqrt 2)))))
278.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
49.0ms
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
27.0ms
(* (* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (cbrt (cbrt (sqrt 2))) (cbrt (cbrt (sqrt 2))))) (cbrt (cbrt (sqrt 2))))

simplify3.2s

Counts
42 → 83
Calls
42 calls:
Slowest
372.0ms
(* (* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (cbrt (cbrt (sqrt 2))) (cbrt (cbrt (sqrt 2))))) (cbrt (cbrt (sqrt 2))))
343.0ms
(* (* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (cbrt (cbrt (sqrt 2))) (cbrt (cbrt (sqrt 2))))) (cbrt (cbrt (sqrt 2))))
338.0ms
(* (* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (cbrt (cbrt (sqrt 2))) (cbrt (cbrt (sqrt 2))))) (cbrt (cbrt (sqrt 2))))
232.0ms
(* (* (* (* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2))))) (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2))))) (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (cbrt (sqrt 2)))
219.0ms
(+ (log (* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* (cbrt (cbrt (sqrt 2))) (cbrt (cbrt (sqrt 2)))))) (log (cbrt (cbrt (sqrt 2)))))

prune1.3s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 3.9b

regimes372.0ms

Accuracy

75.1% (6.0b remaining)

Error of 9.8b against oracle of 3.8b and baseline of 27.8b

bsearch1.1s

end22.0ms

sample8.8s

Algorithm
intervals
Results
2.5s10278×body80nan
1.2s5110×body80valid
994.0ms608×body1280nan
932.0ms886×body1280valid
914.0ms1063×body640valid
572.0ms620×body640nan
560.0ms311×body160valid
461.0ms630×body320valid
176.0ms333×body320nan
69.0ms199×body160nan