Average Error: 0.3 → 0.3
Time: 42.4s
Precision: 64
Internal Precision: 128
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\left(\log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right) + \left(\log z + \log \left(\sqrt[3]{y + x}\right)\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

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Your Program's Arguments

Results

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Derivation

  1. Initial program 0.3

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.3

    \[\leadsto \left(\left(\log \color{blue}{\left(\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}\right)} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  4. Applied log-prod0.3

    \[\leadsto \left(\left(\color{blue}{\left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \log \left(\sqrt[3]{x + y}\right)\right)} + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  5. Applied associate-+l+0.3

    \[\leadsto \left(\color{blue}{\left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \log t\]

  6. Final simplification0.3

    \[\leadsto \left(\left(\log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right) + \left(\log z + \log \left(\sqrt[3]{y + x}\right)\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))

Details

Time bar (total: 40.3s)Debug log

sample453.0ms

Algorithm
intervals

simplify414.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

414.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))

prune29.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.3b

localize52.0ms

Local error

Found 4 expressions with local error:

0.5b
(+ (log (+ x y)) (log z))
0.1b
(* (- a 0.5) (log t))
0.1b
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
0.0b
(- (+ (log (+ x y)) (log z)) t)

rewrite152.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

102.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
32.0ms
(- (+ (log (+ x y)) (log z)) t)
9.0ms
(+ (log (+ x y)) (log z))

series543.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

253.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
160.0ms
(- (+ (log (+ x y)) (log z)) t)
70.0ms
(+ (log (+ x y)) (log z))
60.0ms
(* (- a 0.5) (log t))

simplify4.5s

Counts
52 → 102
Calls

52 calls. Slowest were:

394.0ms
(- (+ (* a (log t)) (+ (log z) (log y))) (+ t (* 0.5 (log t))))
379.0ms
(- (+ (* 0.5 (log (/ -1 t))) (* a (log -1))) (+ (* a (log (/ -1 t))) (* 0.5 (log -1))))
345.0ms
(- (* 0.5 (log (/ 1 t))) (+ t (+ (log (/ 1 z)) (+ (log (/ 1 x)) (* a (log (/ 1 t)))))))

prune1.7s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize34.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (+ x y))
0.6b
(cbrt (+ x y))
0.6b
(cbrt (+ x y))
0.4b
(* (cbrt (+ x y)) (cbrt (+ x y)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

6.0ms
(* (cbrt (+ x y)) (cbrt (+ x y)))
1.0ms
(cbrt (+ x y))
1.0ms
(cbrt (+ x y))

series637.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

176.0ms
(cbrt (+ x y))
175.0ms
(cbrt (+ x y))
146.0ms
(* (cbrt (+ x y)) (cbrt (+ x y)))
139.0ms
(cbrt (+ x y))

simplify2.7s

Counts
59 → 95
Calls

59 calls. Slowest were:

562.0ms
(- (+ (pow y 1/3) (* 1/3 (* x (pow (/ 1 (pow y 2)) 1/3)))) (* 1/9 (* (pow x 2) (pow (/ 1 (pow y 5)) 1/3))))
283.0ms
(- (+ (* 2/3 (* x (pow (/ 1 y) 1/3))) (pow y 2/3)) (* 1/9 (* (pow x 2) (pow (/ 1 (pow y 4)) 1/3))))
245.0ms
(- (+ (pow y 1/3) (* 1/3 (* x (pow (/ 1 (pow y 2)) 1/3)))) (* 1/9 (* (pow x 2) (pow (/ 1 (pow y 5)) 1/3))))

prune1.9s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 4 expressions with local error:

0.7b
(* (cbrt (- a 0.5)) (cbrt (- a 0.5)))
0.5b
(+ (log (+ x y)) (log z))
0.3b
(* (cbrt (- a 0.5)) (log t))
0.3b
(cbrt (- a 0.5))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

10.0ms
(* (cbrt (- a 0.5)) (cbrt (- a 0.5)))
9.0ms
(+ (log (+ x y)) (log z))
9.0ms
(* (cbrt (- a 0.5)) (log t))

series316.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

107.0ms
(* (cbrt (- a 0.5)) (log t))
78.0ms
(+ (log (+ x y)) (log z))
75.0ms
(* (cbrt (- a 0.5)) (cbrt (- a 0.5)))
56.0ms
(cbrt (- a 0.5))

simplify3.6s

Counts
72 → 113
Calls

72 calls. Slowest were:

452.0ms
(- (+ (* (log t) (pow -0.5 1/3)) (* 0.2222222222222222 (* (* (pow a 2) (log t)) (pow -0.5 1/3)))) (+ (* 0.6666666666666666 (* (* a (log t)) (pow -0.5 1/3))) (* 1/6 (* (* (pow a 2) (log t)) (pow -32.0 1/3)))))
331.0ms
(- (* (pow (* a -1) 1/3) (cbrt -1)) (+ (* 0.16666666666666666 (* (pow (/ -1 (pow a 2)) 1/3) (cbrt -1))) (* 0.027777777777777776 (* (pow (/ -1 (pow a 5)) 1/3) (cbrt -1)))))
247.0ms
(- (+ (* 0.027777777777777776 (* (pow (/ -1 (pow a 5)) 1/3) (* (log (/ -1 t)) (cbrt -1)))) (+ (* (pow (* a -1) 1/3) (* (cbrt -1) (log -1))) (* 0.16666666666666666 (* (pow (/ -1 (pow a 2)) 1/3) (* (log (/ -1 t)) (cbrt -1)))))) (+ (* (pow (* a -1) 1/3) (* (log (/ -1 t)) (cbrt -1))) (+ (* 0.16666666666666666 (* (pow (/ -1 (pow a 2)) 1/3) (* (cbrt -1) (log -1)))) (* 0.027777777777777776 (* (pow (/ -1 (pow a 5)) 1/3) (* (cbrt -1) (log -1)))))))

prune2.2s

Pruning

12 alts after pruning (10 fresh and 2 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (* (- a 0.5) (log t)))
0.6b
(cbrt (* (- a 0.5) (log t)))
0.6b
(cbrt (* (- a 0.5) (log t)))
0.5b
(+ (log (+ x y)) (log z))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

9.0ms
(+ (log (+ x y)) (log z))
4.0ms
(cbrt (* (- a 0.5) (log t)))
4.0ms
(cbrt (* (- a 0.5) (log t)))

series406.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

121.0ms
(cbrt (* (- a 0.5) (log t)))
112.0ms
(cbrt (* (- a 0.5) (log t)))
87.0ms
(+ (log (+ x y)) (log z))
85.0ms
(cbrt (* (- a 0.5) (log t)))

simplify3.4s

Counts
31 → 75
Calls

31 calls. Slowest were:

284.0ms
(- (pow (* -0.5 (log t)) 1/3) (+ (* 0.4444444444444444 (* (pow a 2) (pow (* (log t) -0.5) 1/3))) (* 0.6666666666666666 (* a (pow (* (log t) -0.5) 1/3)))))
280.0ms
(- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z))))
271.0ms
(- (pow (* -0.5 (log t)) 1/3) (+ (* 0.4444444444444444 (* (pow a 2) (pow (* (log t) -0.5) 1/3))) (* 0.6666666666666666 (* a (pow (* (log t) -0.5) 1/3)))))

prune1.4s

Pruning

12 alts after pruning (10 fresh and 2 done)

Merged error: 0.0b

regimes1.5s

Accuracy

0% (0.2b remaining)

Error of 0.3b against oracle of 0.1b and baseline of 0.3b

bsearch4.0ms

end0.0ms

sample14.5s

Algorithm
intervals