Average Error: 28.1 → 28.2
Time: 55.3s
Precision: 64
Internal Precision: 128
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{y \cdot \left(230661.510616 + \sqrt[3]{y \cdot \left(y \cdot \left(z + x \cdot y\right) + 27464.7644705\right)} \cdot \left(\sqrt[3]{y \cdot \left(y \cdot \left(z + x \cdot y\right) + 27464.7644705\right)} \cdot \sqrt[3]{y \cdot \left(y \cdot \left(z + x \cdot y\right) + 27464.7644705\right)}\right)\right) + t}{y \cdot \left(c + \left(b + y \cdot \left(y + a\right)\right) \cdot y\right) + i}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.1

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

  3. Applied add-cube-cbrt28.2

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

  4. Final simplification28.2

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

Reproduce

herbie shell --seed 2019022 +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: 53.6s)Debug log

sample77.0ms

Algorithm
intervals
Results
48.0ms260×body80valid

simplify175.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
175.0ms
(/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))

prune34.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 25.3b

localize64.0ms

Local error

Found 4 expressions with local error:

25.3b
(/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
0.1b
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)
0.1b
(* (+ (* (+ y a) y) b) y)
0.1b
(* (+ (* x y) z) y)

rewrite79.0ms

Algorithm
rewrite-expression-head
Rules
15×add-cube-cbrt
15×*-un-lft-identity
15×add-sqr-sqrt
9×associate-*l*
9×times-frac
6×add-exp-log
6×associate-*l/
6×add-cbrt-cube
5×flip-+
5×flip3-+
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×associate-/l*
3×*-commutative
3×associate-/r*
2×associate-/l/
2×associate-/r/
1×div-inv
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 78
Calls
4 calls:
Slowest
56.0ms
(/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
10.0ms
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)
7.0ms
(* (+ (* (+ y a) y) b) y)
4.0ms
(* (+ (* x y) z) y)

series420.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
312.0ms
(/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
39.0ms
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)
36.0ms
(* (+ (* (+ y a) y) b) y)
33.0ms
(* (+ (* x y) z) y)

simplify9.6s

Counts
53 → 90
Calls
53 calls:
Slowest
459.0ms
(+ (* (* (+ (* (+ (* (+ y a) y) b) y) c) y) (* (+ (* (+ (* (+ y a) y) b) y) c) y)) (- (* i i) (* (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))
420.0ms
(+ (pow (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) 3) (pow t 3))
413.0ms
(* (+ (pow (* (+ y a) y) 3) (pow b 3)) y)
405.0ms
(/ (* (cbrt (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)) (cbrt (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t))) 1)
400.0ms
(* (cbrt (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)) (cbrt (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)))

prune2.1s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 24.5b

localize74.0ms

Local error

Found 4 expressions with local error:

25.3b
(/ (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)))
2.8b
(fma (fma (+ y a) y b) (* y y) (fma y c i))
1.5b
(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
Rules
12×add-cube-cbrt
12×*-un-lft-identity
12×add-sqr-sqrt
9×times-frac
6×add-exp-log
6×add-cbrt-cube
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×associate-/l*
3×fma-udef
3×associate-/r*
1×div-inv
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 59
Calls
4 calls:
Slowest
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)
0.0ms
(fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616))

series438.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
271.0ms
(/ (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)))
80.0ms
(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)
27.0ms
(fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616))

simplify4.4s

Counts
37 → 71
Calls
37 calls:
Slowest
435.0ms
(/ 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)))))
388.0ms
(/ 1 (sqrt (fma (fma (+ y a) y b) (* y y) (fma y c i))))
318.0ms
(/ (sqrt (fma y (fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616)) t)) 1)
285.0ms
(sqrt (fma y (fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616)) t))
265.0ms
(/ (* (cbrt (fma y (fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616)) t)) (cbrt (fma y (fma (* y y) (fma y x z) (fma y 27464.7644705 230661.510616)) t))) 1)

prune1.6s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 24.4b

localize33.0ms

Local error

Found 4 expressions with local error:

25.3b
(/ (+ (* (+ (* (* (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) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
19.7b
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))
19.7b
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))
19.7b
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))

rewrite171.0ms

Algorithm
rewrite-expression-head
Rules
12×add-cube-cbrt
12×*-un-lft-identity
12×add-sqr-sqrt
9×times-frac
6×add-exp-log
6×associate-*l/
6×add-cbrt-cube
6×cbrt-div
5×flip-+
5×flip3-+
4×add-log-exp
4×log1p-expm1-u
4×pow1
4×expm1-log1p-u
3×associate-/l*
3×pow1/3
3×associate-/r*
3×cbrt-prod
2×associate-/l/
2×associate-/r/
1×div-inv
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 72
Calls
4 calls:
Slowest
147.0ms
(/ (+ (* (+ (* (* (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) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
9.0ms
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))
9.0ms
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))
5.0ms
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))

series951.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
395.0ms
(/ (+ (* (+ (* (* (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) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
189.0ms
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))
187.0ms
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))
180.0ms
(cbrt (* (+ (* (+ (* x y) z) y) 27464.7644705) y))

simplify10.7s

Counts
50 → 84
Calls
50 calls:
Slowest
457.0ms
(+ (* (* (+ (* (+ (* (+ y a) y) b) y) c) y) (* (+ (* (+ (* (+ y a) y) b) y) c) y)) (- (* i i) (* (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))
435.0ms
(/ (cbrt (+ (* (+ (* (* (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)) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
379.0ms
(cbrt (* (+ (pow (* (+ (* x y) z) y) 3) (pow 27464.7644705 3)) y))
361.0ms
(sqrt (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
355.0ms
(cbrt (* (+ (pow (* (+ (* x y) z) y) 3) (pow 27464.7644705 3)) y))

prune2.2s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 24.4b

localize16.0ms

Local error

Found 4 expressions with local error:

25.3b
(/ (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i) (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t))
0.4b
(/ 1 (/ (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i) (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)))
0.1b
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)
0.1b
(* (+ (* (+ y a) y) b) y)

rewrite109.0ms

Algorithm
rewrite-expression-head
Rules
21×add-cube-cbrt
21×*-un-lft-identity
21×add-sqr-sqrt
18×times-frac
18×associate-/r*
9×add-exp-log
6×associate-*l*
6×add-cbrt-cube
5×flip-+
5×associate-/r/
5×flip3-+
5×pow1
4×add-log-exp
4×log1p-expm1-u
4×associate-*l/
4×expm1-log1p-u
3×associate-/l*
3×div-inv
2×associate-/l/
2×*-commutative
2×div-exp
2×frac-2neg
2×clear-num
2×rec-exp
1×inv-pow
1×pow-flip
1×cbrt-undiv
Counts
4 → 95
Calls
4 calls:
Slowest
55.0ms
(/ (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i) (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t))
21.0ms
(/ 1 (/ (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i) (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)))
20.0ms
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)
12.0ms
(* (+ (* (+ y a) y) b) y)

series900.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
440.0ms
(/ (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i) (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t))
392.0ms
(/ 1 (/ (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i) (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)))
34.0ms
(* (+ (* (+ y a) y) b) y)
33.0ms
(* (+ (* (+ (* x y) z) y) 27464.7644705) y)

simplify13.5s

Counts
66 → 107
Calls
66 calls:
Slowest
513.0ms
(/ (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i) (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t))
494.0ms
(- (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
467.0ms
(sqrt (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t))
466.0ms
(+ (* (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)) (- (* t t) (* (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)))
433.0ms
(+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)

prune2.2s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 24.4b

regimes1.2s

Accuracy

0% (0.8b remaining)

Error of 28.2b against oracle of 27.4b and baseline of 28.2b

bsearch4.0ms

end0.0ms

sample2.6s

Algorithm
intervals
Results
1.8s8255×body80valid