Average Error: 42.0 → 9.5
Time: 37.1s
Precision: 64
Internal Precision: 128
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.07394485735048 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot \left(-t\right) + \frac{t}{x \cdot \sqrt{2}} \cdot -2}\\ \mathbf{elif}\;t \le -3.563799924669161 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{elif}\;t \le -1.8401747941686373 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot \left(-t\right) + \frac{t}{x \cdot \sqrt{2}} \cdot -2}\\ \mathbf{elif}\;t \le 8.159553290541015 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -5.07394485735048e+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. Simplified51.1

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

    3. Taylor expanded around -inf 12.0

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

    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. Simplified26.0

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

    3. Taylor expanded around -inf 9.1

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

    4. Simplified4.1

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

    6. Applied add-cube-cbrt4.1

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

    7. Applied associate-*r*4.0

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

    if -1.8401747941686373e-280 < t < 8.159553290541015e+39

    1. Initial program 42.0

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

    2. Simplified42.0

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

    3. Taylor expanded around -inf 19.4

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

    4. Simplified15.4

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

    6. Applied add-sqr-sqrt15.5

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

    7. Applied associate-*r*15.4

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

    if 8.159553290541015e+39 < t

    1. Initial program 43.2

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

    2. Simplified43.2

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

    3. Taylor expanded around inf 4.9

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

    4. Simplified4.9

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.07394485735048 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot \left(-t\right) + \frac{t}{x \cdot \sqrt{2}} \cdot -2}\\ \mathbf{elif}\;t \le -3.563799924669161 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot \left(t \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{elif}\;t \le -1.8401747941686373 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot \left(-t\right) + \frac{t}{x \cdot \sqrt{2}} \cdot -2}\\ \mathbf{elif}\;t \le 8.159553290541015 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{\sqrt{2}}\right) \cdot t + \left(\sqrt{2} \cdot t\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +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: 35.7s)Debug log

sample807.0ms

Algorithm
intervals

simplify719.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

719.0ms
(/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l))))

prune31.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 44.7b

localize87.0ms

Local error

Found 4 expressions with local error:

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

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

9.0ms
(/ (* t (sqrt 2)) (sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))))
6.0ms
(sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l))))
2.0ms
(* t (sqrt 2))

series732.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

332.0ms
(/ (* t (sqrt 2)) (sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))))
247.0ms
(sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l))))
123.0ms
(fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))
31.0ms
(* t (sqrt 2))

simplify11.6s

Counts
39 → 76
Calls

39 calls. Slowest were:

602.0ms
(- (log (* t (sqrt 2))) (log (sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l))))))
593.0ms
(cbrt (sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l)))))
582.0ms
(sqrt (fma (fma (* 2 t) t (* l l)) (/ (+ 1 x) (+ -1 x)) (* l (- l))))

prune1.2s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 3.8b

localize28.0ms

Local error

Found 4 expressions with local error:

30.5b
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
4.4b
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
0.4b
(* t (sqrt 2))
0.1b
(* (* t 4) (/ t x))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

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))))

series328.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

240.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
50.0ms
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
25.0ms
(* t (sqrt 2))
12.0ms
(* (* t 4) (/ t x))

simplify1.3s

Counts
26 → 68
Calls

26 calls. Slowest were:

523.0ms
(* (* (* (* t 4) (* t 4)) (* t 4)) (* (* (/ t x) (/ t x)) (/ t x)))
135.0ms
(+ (* 2 (pow t 2)) (+ (* 2 (/ (pow l 2) x)) (* 4 (/ (pow t 2) x))))
118.0ms
(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))))))

prune1.3s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 3.8b

localize15.0ms

Local error

Found 4 expressions with local error:

30.5b
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
4.4b
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
0.3b
(* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (cbrt (sqrt 2)))
0.1b
(* (* t 4) (/ t x))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

21.0ms
(* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (cbrt (sqrt 2)))
7.0ms
(* (* t 4) (/ t x))
1.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))

series322.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

215.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
64.0ms
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
25.0ms
(* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (cbrt (sqrt 2)))
19.0ms
(* (* t 4) (/ t x))

simplify1.4s

Counts
32 → 75
Calls

32 calls. Slowest were:

436.0ms
(* (* (* (* t 4) (* t 4)) (* t 4)) (* (* (/ t x) (/ t x)) (/ t x)))
195.0ms
(* (* (* (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2)))) (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2))))) (* t (* (cbrt (sqrt 2)) (cbrt (sqrt 2))))) (sqrt 2))
117.0ms
(+ (* 2 (pow t 2)) (+ (* 2 (/ (pow l 2) x)) (* 4 (/ (pow t 2) x))))

prune1.3s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 3.8b

localize11.0ms

Local error

Found 4 expressions with local error:

30.5b
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
4.4b
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
0.3b
(* (* t (sqrt (sqrt 2))) (sqrt (sqrt 2)))
0.2b
(* t (sqrt (sqrt 2)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

7.0ms
(* (* t (sqrt (sqrt 2))) (sqrt (sqrt 2)))
2.0ms
(* t (sqrt (sqrt 2)))
1.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))

series384.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

279.0ms
(sqrt (fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x))))
40.0ms
(* t (sqrt (sqrt 2)))
38.0ms
(fma 2 (fma (/ l x) l (* t t)) (* (* t 4) (/ t x)))
27.0ms
(* (* t (sqrt (sqrt 2))) (sqrt (sqrt 2)))

simplify930.0ms

Counts
31 → 72
Calls

31 calls. Slowest were:

372.0ms
(* (* (* (* t (sqrt (sqrt 2))) (* t (sqrt (sqrt 2)))) (* t (sqrt (sqrt 2)))) (* (* (sqrt (sqrt 2)) (sqrt (sqrt 2))) (sqrt (sqrt 2))))
113.0ms
(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))))))
94.0ms
(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: 3.8b

regimes527.0ms

Accuracy

75.5% (5.7b remaining)

Error of 9.5b against oracle of 3.9b and baseline of 27.0b

bsearch2.4s

end0.0ms

sample9.1s

Algorithm
intervals