Average Error: 28.3 → 28.4
Time: 54.0s
Precision: 64
Internal Precision: 128
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{y \cdot \left(230661.510616 + \sqrt[3]{y \cdot \left(y \cdot \left(z + x \cdot y\right) + 27464.7644705\right)} \cdot \left(\sqrt[3]{y \cdot \left(y \cdot \left(z + x \cdot y\right) + 27464.7644705\right)} \cdot \sqrt[3]{y \cdot \left(y \cdot \left(z + x \cdot y\right) + 27464.7644705\right)}\right)\right) + t}{(\left((\left(y \cdot y\right) \cdot \left(a + y\right) + \left((y \cdot b + c)_*\right))_*\right) \cdot y + i)_*}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

  1. Initial program 28.3

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity28.3

    \[\leadsto \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\color{blue}{1 \cdot \left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i\right)}}\]

  4. Applied associate-/r*28.3

    \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{1}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}\]

  5. Simplified28.3

    \[\leadsto \frac{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{1}}{\color{blue}{(\left((\left(y \cdot y\right) \cdot \left(a + y\right) + \left((y \cdot b + c)_*\right))_*\right) \cdot y + i)_*}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt28.4

    \[\leadsto \frac{\frac{\left(\color{blue}{\left(\sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y} \cdot \sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y}\right) \cdot \sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y}} + 230661.510616\right) \cdot y + t}{1}}{(\left((\left(y \cdot y\right) \cdot \left(a + y\right) + \left((y \cdot b + c)_*\right))_*\right) \cdot y + i)_*}\]

  8. Final simplification28.4

    \[\leadsto \frac{y \cdot \left(230661.510616 + \sqrt[3]{y \cdot \left(y \cdot \left(z + x \cdot y\right) + 27464.7644705\right)} \cdot \left(\sqrt[3]{y \cdot \left(y \cdot \left(z + x \cdot y\right) + 27464.7644705\right)} \cdot \sqrt[3]{y \cdot \left(y \cdot \left(z + x \cdot y\right) + 27464.7644705\right)}\right)\right) + t}{(\left((\left(y \cdot y\right) \cdot \left(a + y\right) + \left((y \cdot b + c)_*\right))_*\right) \cdot y + i)_*}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

Details

Time bar (total: 51.2s)Debug log

sample130.0ms

Algorithm
intervals

simplify176.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

176.0ms
(/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))

prune25.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 30.2b

localize49.0ms

Local error

Found 4 expressions with local error:

30.2b
(/ (fma y (fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616)) t) (fma (fma (+ y a) y b) (* y y) (fma y c i)))
3.0b
(fma (fma (+ y a) y b) (* y y) (fma y c i))
0.3b
(fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616))
0.0b
(fma y (fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616)) t)

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

7.0ms
(/ (fma y (fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616)) t) (fma (fma (+ y a) y b) (* y y) (fma y c i)))
0.0ms
(fma (fma (+ y a) y b) (* y y) (fma y c i))
0.0ms
(fma y (fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616)) t)

series452.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

276.0ms
(/ (fma y (fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616)) t) (fma (fma (+ y a) y b) (* y y) (fma y c i)))
69.0ms
(fma (fma (+ y a) y b) (* y y) (fma y c i))
60.0ms
(fma y (fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616)) t)
47.0ms
(fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616))

simplify4.6s

Counts
37 → 71
Calls

37 calls. Slowest were:

451.0ms
(/ 1 (* (cbrt (fma (fma (+ y a) y b) (* y y) (fma y c i))) (cbrt (fma (fma (+ y a) y b) (* y y) (fma y c i)))))
447.0ms
(/ 1 (sqrt (fma (fma (+ y a) y b) (* y y) (fma y c i))))
307.0ms
(/ (sqrt (fma y (fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616)) t)) 1)

prune1.7s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 29.6b

localize85.0ms

Local error

Found 4 expressions with local error:

30.2b
(/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
0.1b
(* (+ (* (+ y a) y) b) y)
0.1b
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)
0.1b
(* (+ (* (+ (* (+ y a) y) b) y) c) y)

rewrite138.0ms

Algorithm
rewrite-expression-head
Counts
4 → 78
Calls

4 calls. Slowest were:

79.0ms
(/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
21.0ms
(* (+ (* (+ y a) y) b) y)
20.0ms
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)

series412.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

307.0ms
(/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
46.0ms
(* (+ (* (+ (* (+ y a) y) b) y) c) y)
36.0ms
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)
23.0ms
(* (+ (* (+ y a) y) b) y)

simplify11.3s

Counts
53 → 90
Calls

53 calls. Slowest were:

824.0ms
(* (+ (pow (* (+ y a) y) 3) (pow b 3)) y)
508.0ms
(+ (pow (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) 3) (pow t 3))
460.0ms
(* (+ (pow (* (+ (* (+ y a) y) b) y) 3) (pow c 3)) y)

prune2.1s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 29.6b

localize43.0ms

Local error

Found 4 expressions with local error:

30.2b
(/ (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) 1) (fma (fma (* y y) (+ a y) (fma y b c)) y i))
4.3b
(fma (* y y) (+ a y) (fma y b c))
0.1b
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)
0.1b
(* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)

rewrite78.0ms

Algorithm
rewrite-expression-head
Counts
4 → 74
Calls

4 calls. Slowest were:

34.0ms
(* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
32.0ms
(/ (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) 1) (fma (fma (* y y) (+ a y) (fma y b c)) y i))
10.0ms
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)

series459.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

303.0ms
(/ (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) 1) (fma (fma (* y y) (+ a y) (fma y b c)) y i))
62.0ms
(* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
58.0ms
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)
36.0ms
(fma (* y y) (+ a y) (fma y b c))

simplify9.1s

Counts
53 → 86
Calls

53 calls. Slowest were:

463.0ms
(* (+ (pow (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 3) (pow 230661.510616 3)) y)
452.0ms
(sqrt (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) 1))
427.0ms
(* (- (* (* (+ (* (+ (* x y) z) y) 27464.7644705) y) (* (+ (* (+ (* x y) z) y) 27464.7644705) y)) (* 230661.510616 230661.510616)) y)

prune2.2s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 29.6b

localize53.0ms

Local error

Found 4 expressions with local error:

30.2b
(/ (/ (+ (* (+ (* (* (cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y)) (cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))) (cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))) 230661.510616) y) t) 1) (fma (fma (* y y) (+ a y) (fma y b c)) y i))
21.1b
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))
21.1b
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))
21.1b
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))

rewrite61.0ms

Algorithm
rewrite-expression-head
Counts
4 → 73
Calls

4 calls. Slowest were:

28.0ms
(/ (/ (+ (* (+ (* (* (cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y)) (cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))) (cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))) 230661.510616) y) t) 1) (fma (fma (* y y) (+ a y) (fma y b c)) y i))
11.0ms
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))
11.0ms
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))

series967.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

421.0ms
(/ (/ (+ (* (+ (* (* (cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y)) (cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))) (cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))) 230661.510616) y) t) 1) (fma (fma (* y y) (+ a y) (fma y b c)) y i))
212.0ms
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))
171.0ms
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))
162.0ms
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))

simplify10.3s

Counts
54 → 85
Calls

54 calls. Slowest were:

714.0ms
(cbrt (* (- (* (* (+ (* x y) z) y) (* (+ (* x y) z) y)) (* 27464.7644705 27464.7644705)) y))
421.0ms
(cbrt (* (+ (pow (* (+ (* x y) z) y) 3) (pow 27464.7644705 3)) y))
383.0ms
(cbrt (* (+ (pow (* (+ (* x y) z) y) 3) (pow 27464.7644705 3)) y))

prune2.1s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 29.6b

regimes1.5s

Accuracy

0% (0.8b remaining)

Error of 28.4b against oracle of 27.6b and baseline of 28.4b

bsearch4.0ms

end0.0ms

sample3.1s

Algorithm
intervals