Average Error: 0.3 → 0.3
Time: 31.8s
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 2019022 
(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: 30.2s)Debug log

sample645.0ms

Algorithm
intervals
Results
528.0ms1734×body80nan
45.0ms256×body80valid

simplify287.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
287.0ms
(+ (- (+ (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

localize56.0ms

Local error

Found 3 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))

rewrite135.0ms

Algorithm
rewrite-expression-head
Rules
9×*-un-lft-identity
6×add-cube-cbrt
6×associate-*l/
6×associate-+l+
6×pow1
6×add-sqr-sqrt
5×add-log-exp
5×flip--
5×add-exp-log
5×add-cbrt-cube
5×flip3--
4×frac-add
3×flip-+
3×associate-+l-
3×associate-*l*
3×associate-*r*
3×flip3-+
3×log-prod
2×distribute-lft-out
2×sum-log
2×log-div
2×+-commutative
1×cbrt-unprod
1×*-commutative
1×log-pow
1×associate--l+
1×prod-exp
1×pow-prod-down
1×sub-neg
Counts
3 → 56
Calls
3 calls:
Slowest
117.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
8.0ms
(+ (log (+ x y)) (log z))
8.0ms
(* (- a 0.5) (log t))

series359.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
206.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
98.0ms
(+ (log (+ x y)) (log z))
55.0ms
(* (- a 0.5) (log t))

simplify2.9s

Counts
40 → 65
Calls
40 calls:
Slowest
370.0ms
(- (+ (* a (log t)) (+ (log z) (log y))) (+ t (* 0.5 (log t))))
242.0ms
(- (* 0.5 (log (/ 1 t))) (+ t (+ (log (/ 1 z)) (+ (log (/ 1 x)) (* a (log (/ 1 t)))))))
213.0ms
(- (+ (* 0.5 (log (/ -1 t))) (* a (log -1))) (+ (* a (log (/ -1 t))) (* 0.5 (log -1))))
197.0ms
(+ (* (- (pow (+ (log (+ x y)) (log z)) 3) (pow t 3)) (+ (* a a) (+ (* 0.5 0.5) (* a 0.5)))) (* (+ (* (+ (log (+ x y)) (log z)) (+ (log (+ x y)) (log z))) (+ (* t t) (* (+ (log (+ x y)) (log z)) t))) (* (- (pow a 3) (pow 0.5 3)) (log t))))
196.0ms
(- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z))))

prune930.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize28.0ms

Local error

Found 4 expressions with local error:

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

rewrite7.0ms

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))
0.0ms
(cbrt (+ x y))
0.0ms
(cbrt (+ x y))

series534.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
161.0ms
(* (cbrt (+ x y)) (cbrt (+ x y)))
126.0ms
(cbrt (+ x y))
125.0ms
(cbrt (+ x y))
121.0ms
(cbrt (+ x y))

simplify1.4s

Counts
59 → 87
Calls
59 calls:
Slowest
210.0ms
(- (+ (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
(- (+ (* 2/3 (* x (pow (/ 1 y) 1/3))) (pow y 2/3)) (* 1/9 (* (pow x 2) (pow (/ 1 (pow y 4)) 1/3))))
153.0ms
(- (+ (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))))
146.0ms
(- (+ (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))))
80.0ms
(* (cbrt (- (* x x) (* y y))) (cbrt (- (* x x) (* y y))))

prune1.8s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
14×*-un-lft-identity
12×cbrt-prod
11×add-cube-cbrt
11×add-sqr-sqrt
10×pow1
6×add-exp-log
6×associate-*l*
6×associate-*r*
6×pow1/3
4×add-log-exp
4×add-cbrt-cube
3×log-prod
3×associate-+l+
2×distribute-lft-out
2×flip-+
2×associate-+l-
2×pow-prod-up
2×pow-prod-down
2×flip3-+
2×log-div
1×cbrt-unprod
1×*-commutative
1×log-pow
1×prod-exp
1×pow-plus
1×sum-log
1×+-commutative
1×pow2
Counts
4 → 68
Calls
4 calls:
Slowest
5.0ms
(+ (log (+ x y)) (log z))
4.0ms
(* (cbrt (log t)) (cbrt (log t)))
0.0ms
(cbrt (log t))
0.0ms
(cbrt (log t))

series1.2s

Counts
4 → 12
Calls
4 calls:
Slowest
384.0ms
(* (cbrt (log t)) (cbrt (log t)))
346.0ms
(cbrt (log t))
329.0ms
(cbrt (log t))
91.0ms
(+ (log (+ x y)) (log z))

simplify738.0ms

Counts
46 → 80
Calls
46 calls:
Slowest
340.0ms
(pow (pow (- (log -1) (log (/ -1 t))) 2) 1/3)
201.0ms
(- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z))))
38.0ms
(- (+ (log (/ 1 z)) (log (/ 1 x))))
36.0ms
(pow (- (log -1) (log (/ -1 t))) 1/3)
32.0ms
(log (- (* x x) (* y y)))

prune1.4s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0b

localize23.0ms

Local error

Found 4 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
11×*-un-lft-identity
10×pow1
8×add-cube-cbrt
8×add-sqr-sqrt
6×add-exp-log
6×add-cbrt-cube
4×add-log-exp
4×cbrt-prod
3×associate-*l*
3×associate-*r*
3×log-prod
3×associate-+l+
2×distribute-lft-out
2×flip-+
2×associate-+l-
2×pow1/3
2×flip3-+
2×log-div
1×cbrt-unprod
1×pow-prod-up
1×*-commutative
1×log-pow
1×prod-exp
1×pow-prod-down
1×pow-plus
1×sum-log
1×+-commutative
1×pow2
Counts
4 → 58
Calls
4 calls:
Slowest
4.0ms
(+ (log (+ x y)) (log z))
3.0ms
(* (log t) (log t))
2.0ms
(cbrt (* (log t) (log t)))
0.0ms
(cbrt (log t))

series949.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
362.0ms
(cbrt (* (log t) (log t)))
331.0ms
(cbrt (log t))
185.0ms
(* (log t) (log t))
71.0ms
(+ (log (+ x y)) (log z))

simplify990.0ms

Counts
35 → 70
Calls
35 calls:
Slowest
332.0ms
(pow (pow (- (log -1) (log (/ -1 t))) 2) 1/3)
252.0ms
(pow (- (log -1) (log (/ -1 t))) 2)
230.0ms
(- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z))))
31.0ms
(* (* (* (log t) (log t)) (log t)) (* (* (log t) (log t)) (log t)))
25.0ms
(log (- (* x x) (* y y)))

prune1.2s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0b

regimes1.3s

Accuracy

0% (0.2b remaining)

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

bsearch4.0ms

end0.0ms

sample13.3s

Algorithm
intervals
Results
10.1s56306×body80nan
1.5s8025×body80valid