Average Error: 0.3 → 0.3
Time: 44.5s
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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 2019010 
(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: 42.4s)Debug log

sample476.0ms

Algorithm
intervals

simplify255.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
254.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize43.0ms

Local error

Found 4 expressions with local error:

0.6b
(+ (log (+ x y)) (log z))
0.1b
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
0.1b
(* (- a 0.5) (log t))
0.0b
(- (+ (log (+ x y)) (log z)) t)

rewrite137.0ms

Algorithm
rewrite-expression-head
Rules
10×*-un-lft-identity
7×add-cube-cbrt
7×pow1
7×add-sqr-sqrt
6×add-log-exp
6×flip--
6×add-exp-log
6×associate-*l/
6×add-cbrt-cube
6×flip3--
6×associate-+l+
5×associate-+l-
4×flip-+
4×frac-add
4×flip3-+
4×log-div
3×associate-*l*
3×associate-*r*
3×log-prod
2×distribute-lft-out
2×associate--l+
2×sub-neg
2×sum-log
2×+-commutative
2×associate--l-
1×cbrt-unprod
1×*-commutative
1×log-pow
1×prod-exp
1×pow-prod-down
Counts
4 → 70
Calls
4 calls:
Slowest
92.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
30.0ms
(- (+ (log (+ x y)) (log z)) t)
9.0ms
(+ (log (+ x y)) (log z))
4.0ms
(* (- a 0.5) (log t))

series519.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
198.0ms
(- (+ (log (+ x y)) (log z)) t)
186.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
74.0ms
(* (- a 0.5) (log t))
62.0ms
(+ (log (+ x y)) (log z))

simplify3.4s

Counts
46 → 82
Calls
46 calls:
Slowest
353.0ms
(- (+ (* a (log t)) (+ (log z) (log y))) (+ t (* 0.5 (log t))))
322.0ms
(- (+ (* 0.5 (log (/ -1 t))) (* a (log -1))) (+ (* a (log (/ -1 t))) (* 0.5 (log -1))))
250.0ms
(- (* 0.5 (log (/ 1 t))) (+ t (+ (log (/ 1 z)) (+ (log (/ 1 x)) (* a (log (/ 1 t)))))))
245.0ms
(- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z))))
193.0ms
(* (* (* (- a 0.5) (- a 0.5)) (- a 0.5)) (* (* (log t) (log t)) (log t)))

prune1.3s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize48.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (+ x y))
0.6b
(cbrt (+ x y))
0.6b
(cbrt (+ x y))
0.5b
(* (cbrt (+ x y)) (cbrt (+ x y)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
18×cbrt-div
15×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×flip-+
9×flip3-+
9×pow1
7×pow1/3
6×add-exp-log
6×associate-*l*
6×associate-*r*
4×add-log-exp
4×add-cbrt-cube
4×frac-times
2×pow-prod-up
2×associate-*r/
2×associate-*l/
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 75
Calls
4 calls:
Slowest
5.0ms
(* (cbrt (+ x y)) (cbrt (+ x y)))
1.0ms
(cbrt (+ x y))
1.0ms
(cbrt (+ x y))
1.0ms
(cbrt (+ x y))

series626.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
175.0ms
(* (cbrt (+ x y)) (cbrt (+ x y)))
151.0ms
(cbrt (+ x y))
151.0ms
(cbrt (+ x y))
149.0ms
(cbrt (+ x y))

simplify1.5s

Counts
59 → 87
Calls
59 calls:
Slowest
205.0ms
(- (+ (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))))
192.0ms
(- (+ (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))))
164.0ms
(- (+ (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))))
118.0ms
(- (+ (* 2/3 (* x (pow (/ 1 y) 1/3))) (pow y 2/3)) (* 1/9 (* (pow x 2) (pow (/ 1 (pow y 4)) 1/3))))
78.0ms
(* (cbrt (- (* x x) (* y y))) (cbrt (- (* x x) (* y y))))

prune1.7s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.0b

localize36.0ms

Local error

Found 4 expressions with local error:

0.6b
(+ (log (+ x y)) (log z))
0.6b
(+ (log (+ x y)) (log z))
0.6b
(+ (log (+ x y)) (log z))
0.6b
(cbrt (+ (log (+ x y)) (log z)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
20×*-un-lft-identity
9×log-prod
9×associate-+l+
8×distribute-lft-out
8×add-cube-cbrt
8×pow1
8×add-sqr-sqrt
7×flip-+
7×flip3-+
6×associate-+l-
6×log-div
5×cbrt-prod
4×add-log-exp
4×log-pow
4×add-exp-log
4×add-cbrt-cube
3×sum-log
3×+-commutative
2×cbrt-div
1×pow1/3
Counts
4 → 69
Calls
4 calls:
Slowest
9.0ms
(+ (log (+ x y)) (log z))
8.0ms
(+ (log (+ x y)) (log z))
8.0ms
(+ (log (+ x y)) (log z))
5.0ms
(cbrt (+ (log (+ x y)) (log z)))

series498.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
270.0ms
(cbrt (+ (log (+ x y)) (log z)))
84.0ms
(+ (log (+ x y)) (log z))
74.0ms
(+ (log (+ x y)) (log z))
70.0ms
(+ (log (+ x y)) (log z))

simplify1.3s

Counts
43 → 81
Calls
43 calls:
Slowest
253.0ms
(- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z))))
242.0ms
(pow (- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z)))) 1/3)
204.0ms
(- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z))))
180.0ms
(- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z))))
51.0ms
(cbrt (- (* (log (+ x y)) (log (+ x y))) (* (log z) (log z))))

prune1.7s

Pruning

12 alts after pruning (10 fresh and 2 done)

Merged error: 0.0b

localize29.0ms

Local error

Found 4 expressions with local error:

0.7b
(* (cbrt (- a 0.5)) (cbrt (- a 0.5)))
0.6b
(+ (log (+ x y)) (log z))
0.4b
(* (cbrt (- a 0.5)) (log t))
0.3b
(* (* (cbrt (- a 0.5)) (cbrt (- a 0.5))) (* (cbrt (- a 0.5)) (log t)))

rewrite79.0ms

Algorithm
rewrite-expression-head
Rules
68×cbrt-div
36×associate-*l/
34×flip--
34×flip3--
32×frac-times
24×pow1
20×add-exp-log
15×*-un-lft-identity
13×associate-*l*
12×add-cube-cbrt
12×add-sqr-sqrt
11×pow-prod-down
11×add-cbrt-cube
10×cbrt-unprod
10×associate-*r/
10×prod-exp
10×associate-*r*
9×cbrt-prod
4×add-log-exp
4×pow1/3
3×*-commutative
3×log-prod
3×associate-+l+
2×distribute-lft-out
2×flip-+
2×associate-+l-
2×pow-prod-up
2×flip3-+
2×log-div
1×log-pow
1×pow-plus
1×sum-log
1×+-commutative
1×pow2
Counts
4 → 124
Calls
4 calls:
Slowest
47.0ms
(* (* (cbrt (- a 0.5)) (cbrt (- a 0.5))) (* (cbrt (- a 0.5)) (log t)))
10.0ms
(* (cbrt (- a 0.5)) (cbrt (- a 0.5)))
9.0ms
(* (cbrt (- a 0.5)) (log t))
9.0ms
(+ (log (+ x y)) (log z))

series354.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
138.0ms
(* (cbrt (- a 0.5)) (log t))
88.0ms
(* (* (cbrt (- a 0.5)) (cbrt (- a 0.5))) (* (cbrt (- a 0.5)) (log t)))
67.0ms
(* (cbrt (- a 0.5)) (cbrt (- a 0.5)))
61.0ms
(+ (log (+ x y)) (log z))

simplify8.2s

Counts
121 → 136
Calls
121 calls:
Slowest
631.0ms
(* (* (* (* (cbrt (- a 0.5)) (cbrt (- a 0.5))) (* (cbrt (- a 0.5)) (cbrt (- a 0.5)))) (* (cbrt (- a 0.5)) (cbrt (- a 0.5)))) (* (- a 0.5) (* (* (log t) (log t)) (log t))))
438.0ms
(* (* (* (* (cbrt (- a 0.5)) (cbrt (- a 0.5))) (* (cbrt (- a 0.5)) (cbrt (- a 0.5)))) (* (cbrt (- a 0.5)) (cbrt (- a 0.5)))) (* (* (* (cbrt (- a 0.5)) (log t)) (* (cbrt (- a 0.5)) (log t))) (* (cbrt (- a 0.5)) (log t))))
366.0ms
(- (* (pow (pow a 2) 1/3) (pow (cbrt -1) 2)) (+ (* 0.027777777777777776 (* (pow (/ 1 (pow a 4)) 1/3) (pow (cbrt -1) 2))) (* 0.3333333333333333 (* (pow (/ 1 a) 1/3) (pow (cbrt -1) 2)))))
308.0ms
(- (+ (* (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)))))
306.0ms
(* (* (- a 0.5) (- a 0.5)) (* (* (* (cbrt (- a 0.5)) (log t)) (* (cbrt (- a 0.5)) (log t))) (* (cbrt (- a 0.5)) (log t))))

prune2.9s

Pruning

13 alts after pruning (10 fresh and 3 done)

Merged error: 0.0b

regimes2.2s

Accuracy

0% (0.2b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch2.0ms

end0.0ms

sample15.1s

Algorithm
intervals