Average Error: 0.3 → 0.3
Time: 39.4s
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

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 2019016 
(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: 37.4s)Debug log

sample539.0ms

Algorithm
intervals
Results
2146×(pre true 80)
1888×(body nan 80)
258×(body real 80)

simplify346.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
345.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))

prune19.0ms

Pruning

1 alts after pruning (1 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))
0.2b
(* (- 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)

rewrite164.0ms

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
123.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
25.0ms
(- (+ (log (+ x y)) (log z)) t)
8.0ms
(+ (log (+ x y)) (log z))
7.0ms
(* (- a 0.5) (log t))

series596.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
247.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
205.0ms
(- (+ (log (+ x y)) (log z)) t)
75.0ms
(* (- a 0.5) (log t))
69.0ms
(+ (log (+ x y)) (log z))

simplify3.4s

Counts
46 → 82
Calls
46 calls:
Slowest
371.0ms
(- (+ (* a (log t)) (+ (log z) (log y))) (+ t (* 0.5 (log t))))
282.0ms
(- (* 0.5 (log (/ 1 t))) (+ t (+ (log (/ 1 z)) (+ (log (/ 1 x)) (* a (log (/ 1 t)))))))
274.0ms
(- (+ (* 0.5 (log (/ -1 t))) (* a (log -1))) (+ (* a (log (/ -1 t))) (* 0.5 (log -1))))
269.0ms
(- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z))))
198.0ms
(+ (* (- (* (+ (log (+ x y)) (log z)) (+ (log (+ x y)) (log z))) (* t t)) (+ (* a a) (+ (* 0.5 0.5) (* a 0.5)))) (* (+ (+ (log (+ x y)) (log z)) t) (* (- (pow a 3) (pow 0.5 3)) (log t))))

prune1.3s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize40.0ms

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)))

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

series619.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
171.0ms
(* (cbrt (+ x y)) (cbrt (+ x y)))
166.0ms
(cbrt (+ x y))
151.0ms
(cbrt (+ x y))
131.0ms
(cbrt (+ x y))

simplify1.6s

Counts
59 → 87
Calls
59 calls:
Slowest
211.0ms
(- (+ (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))))
190.0ms
(- (+ (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))))
184.0ms
(- (+ (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))))
169.0ms
(- (+ (* 2/3 (* x (pow (/ 1 y) 1/3))) (pow y 2/3)) (* 1/9 (* (pow x 2) (pow (/ 1 (pow y 4)) 1/3))))
80.0ms
(* (cbrt (- (* x x) (* y y))) (cbrt (+ (pow x 3) (pow y 3))))

prune1.9s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

localize31.0ms

Local error

Found 4 expressions with local error:

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

rewrite41.0ms

Algorithm
rewrite-expression-head
Rules
26×*-un-lft-identity
25×cbrt-prod
18×cbrt-div
14×pow1
11×add-cube-cbrt
11×add-sqr-sqrt
10×distribute-lft-out
9×flip-+
9×flip3-+
8×associate-*l*
8×associate-*r*
7×pow1/3
6×add-exp-log
5×log-pow
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 → 85
Calls
4 calls:
Slowest
23.0ms
(* (cbrt (+ (log (+ x y)) (log z))) (cbrt (+ (log (+ x y)) (log z))))
6.0ms
(cbrt (+ (log (+ x y)) (log z)))
5.0ms
(cbrt (+ (log (+ x y)) (log z)))
5.0ms
(cbrt (+ (log (+ x y)) (log z)))

series2.1s

Counts
4 → 12
Calls
4 calls:
Slowest
1.1s
(* (cbrt (+ (log (+ x y)) (log z))) (cbrt (+ (log (+ x y)) (log z))))
315.0ms
(cbrt (+ (log (+ x y)) (log z)))
307.0ms
(cbrt (+ (log (+ x y)) (log z)))
304.0ms
(cbrt (+ (log (+ x y)) (log z)))

simplify2.2s

Counts
69 → 97
Calls
69 calls:
Slowest
239.0ms
(pow (- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z)))) 1/3)
227.0ms
(pow (- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z)))) 1/3)
195.0ms
(pow (- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z)))) 1/3)
159.0ms
(pow (pow (+ (log (/ 1 z)) (log (/ 1 x))) 2) 1/3)
139.0ms
(pow (pow (- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z)))) 2) 1/3)

prune1.8s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
11×*-un-lft-identity
9×cbrt-prod
8×add-cube-cbrt
8×add-sqr-sqrt
5×pow1
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
3×pow1/3
3×log-prod
3×associate-+l+
2×distribute-lft-out
2×flip-+
2×associate-+l-
2×flip3-+
2×log-div
1×log-pow
1×sum-log
1×+-commutative
Counts
4 → 51
Calls
4 calls:
Slowest
5.0ms
(+ (log (+ x y)) (log z))
1.0ms
(cbrt (log t))
1.0ms
(cbrt (log t))
1.0ms
(cbrt (log t))

series1.1s

Counts
4 → 12
Calls
4 calls:
Slowest
347.0ms
(cbrt (log t))
341.0ms
(cbrt (log t))
327.0ms
(cbrt (log t))
61.0ms
(+ (log (+ x y)) (log z))

simplify453.0ms

Counts
29 → 63
Calls
29 calls:
Slowest
216.0ms
(- (* 2 (log -1)) (+ (log (/ -1 x)) (log (/ -1 z))))
41.0ms
(- (+ (log (/ 1 z)) (log (/ 1 x))))
39.0ms
(pow (- (log -1) (log (/ -1 t))) 1/3)
37.0ms
(log (- (* x x) (* y y)))
33.0ms
(pow (- (log -1) (log (/ -1 t))) 1/3)

prune1.1s

Pruning

12 alts after pruning (10 fresh and 2 done)

Merged error: 0.0b

regimes1.7s

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms

end0.0ms

sample16.2s

Algorithm
intervals
Results
64159×(pre true 80)
56119×(body nan 80)
8040×(body real 80)