Average Error: 0.3 → 0.3
Time: 42.0s
Precision: 64
Internal Precision: 128
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\log \left(\sqrt[3]{x + y}\right) + \left((\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_* - t\right)\right) + \log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right)\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

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. Simplified0.3

    \[\leadsto \color{blue}{\log \left(y + x\right) + \left((\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_* - t\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+l+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019016 +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: 40.2s)Debug log

sample864.0ms

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

simplify407.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
406.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))

prune35.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize33.0ms

Local error

Found 3 expressions with local error:

0.4b
(fma (- a 0.5) (log t) (log z))
0.1b
(+ (log (+ y x)) (- (fma (- a 0.5) (log t) (log z)) t))
0.0b
(- (fma (- a 0.5) (log t) (log z)) t)

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
9×*-un-lft-identity
6×add-cube-cbrt
6×add-sqr-sqrt
5×pow1
4×add-log-exp
4×fma-def
3×log1p-expm1-u
3×fma-udef
3×add-exp-log
3×fma-neg
3×add-cbrt-cube
3×log-prod
3×associate-+r+
3×associate-+l+
3×expm1-log1p-u
2×distribute-lft-out
2×flip-+
2×associate-+l-
2×log-pow
2×associate--l+
2×flip3-+
2×sub-neg
2×log-div
1×flip--
1×flip3--
1×sum-log
1×associate-+r-
1×+-commutative
Counts
3 → 55
Calls
3 calls:
Slowest
9.0ms
(+ (log (+ y x)) (- (fma (- a 0.5) (log t) (log z)) t))
3.0ms
(- (fma (- a 0.5) (log t) (log z)) t)
0.0ms
(fma (- a 0.5) (log t) (log z))

series312.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
183.0ms
(+ (log (+ y x)) (- (fma (- a 0.5) (log t) (log z)) t))
84.0ms
(fma (- a 0.5) (log t) (log z))
45.0ms
(- (fma (- a 0.5) (log t) (log z)) t)

simplify3.2s

Counts
25 → 64
Calls
25 calls:
Slowest
448.0ms
(- (* 0.5 (log (/ 1 t))) (+ t (+ (log (/ 1 z)) (* a (log (/ 1 t))))))
431.0ms
(- (* 0.5 (log (/ 1 t))) (+ (log (/ 1 z)) (* a (log (/ 1 t)))))
420.0ms
(- (+ (* a (log t)) (+ (log z) (log x))) (+ t (* 0.5 (log t))))
408.0ms
(- (+ (* 0.5 (log (/ -1 t))) (+ (* a (log -1)) (* 0.5 (log -1)))) (+ t (+ (* a (log (/ -1 t))) (log (/ -1 z)))))
295.0ms
(- (* 0.5 (log (/ 1 t))) (+ t (+ (log (/ 1 z)) (+ (* a (log (/ 1 t))) (log (/ 1 y))))))

prune972.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.1b

localize40.0ms

Local error

Found 4 expressions with local error:

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

rewrite15.0ms

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×log1p-expm1-u
4×add-cbrt-cube
4×frac-times
4×expm1-log1p-u
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 → 83
Calls
4 calls:
Slowest
10.0ms
(* (cbrt (+ y x)) (cbrt (+ y x)))
1.0ms
(cbrt (+ y x))
1.0ms
(cbrt (+ y x))
1.0ms
(cbrt (+ y x))

series627.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
182.0ms
(* (cbrt (+ y x)) (cbrt (+ y x)))
156.0ms
(cbrt (+ y x))
145.0ms
(cbrt (+ y x))
145.0ms
(cbrt (+ y x))

simplify2.1s

Counts
59 → 95
Calls
59 calls:
Slowest
246.0ms
(- (+ (pow x 1/3) (* 1/3 (* (pow (/ 1 (pow x 2)) 1/3) y))) (* 1/9 (* (pow (/ 1 (pow x 5)) 1/3) (pow y 2))))
223.0ms
(- (+ (pow x 1/3) (* 1/3 (* (pow (/ 1 (pow x 2)) 1/3) y))) (* 1/9 (* (pow (/ 1 (pow x 5)) 1/3) (pow y 2))))
212.0ms
(- (+ (pow x 1/3) (* 1/3 (* (pow (/ 1 (pow x 2)) 1/3) y))) (* 1/9 (* (pow (/ 1 (pow x 5)) 1/3) (pow y 2))))
205.0ms
(- (+ (* 2/3 (* (pow (/ 1 x) 1/3) y)) (pow x 2/3)) (* 1/9 (* (pow (/ 1 (pow x 4)) 1/3) (pow y 2))))
117.0ms
(* (cbrt (- (* y y) (* x x))) (cbrt (+ (pow y 3) (pow x 3))))

prune2.0s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.1b

localize12.0ms

Local error

Found 4 expressions with local error:

0.4b
(fma (- a 0.5) (log t) (log z))
0.1b
(+ (log (sqrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t))
0.1b
(+ (log (sqrt (+ y x))) (+ (log (sqrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t)))
0.0b
(log (sqrt (+ y x)))

rewrite48.0ms

Algorithm
rewrite-expression-head
Rules
26×*-un-lft-identity
18×log-prod
13×pow1
12×distribute-lft-out
12×add-cube-cbrt
12×log-pow
12×associate-+l+
12×add-sqr-sqrt
10×fma-def
9×sqrt-prod
8×sqrt-div
8×log-div
7×add-log-exp
6×flip-+
6×associate-+l-
6×flip3-+
5×add-exp-log
5×associate-+r-
4×log1p-expm1-u
4×add-cbrt-cube
4×sum-log
4×associate-+r+
4×expm1-log1p-u
3×pow1/2
2×fma-udef
2×+-commutative
1×associate--l+
1×rem-log-exp
1×sub-neg
Counts
4 → 99
Calls
4 calls:
Slowest
31.0ms
(+ (log (sqrt (+ y x))) (+ (log (sqrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t)))
14.0ms
(+ (log (sqrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t))
1.0ms
(log (sqrt (+ y x)))
0.0ms
(fma (- a 0.5) (log t) (log z))

series569.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
222.0ms
(+ (log (sqrt (+ y x))) (+ (log (sqrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t)))
189.0ms
(+ (log (sqrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t))
97.0ms
(fma (- a 0.5) (log t) (log z))
60.0ms
(log (sqrt (+ y x)))

simplify4.7s

Counts
58 → 111
Calls
58 calls:
Slowest
468.0ms
(- (* 0.5 (log (/ 1 t))) (+ (log (/ 1 z)) (* a (log (/ 1 t)))))
365.0ms
(+ (log (sqrt (+ y x))) (+ (log (sqrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t)))
339.0ms
(+ (log (sqrt (+ y x))) (+ (log (sqrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t)))
327.0ms
(* (sqrt (+ y x)) (exp (+ (log (sqrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t))))
325.0ms
(+ (log (sqrt (+ y x))) (+ (log (sqrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t)))

prune1.8s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.1b

localize32.0ms

Local error

Found 4 expressions with local error:

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

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
30×cbrt-prod
18×sqrt-div
18×cbrt-div
16×add-cube-cbrt
16×*-un-lft-identity
16×add-sqr-sqrt
15×sqrt-prod
9×flip-+
9×associate-*l*
9×associate-*r*
9×flip3-+
9×pow1
7×pow1/3
6×add-exp-log
4×add-log-exp
4×log1p-expm1-u
4×add-cbrt-cube
4×frac-times
4×expm1-log1p-u
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 → 98
Calls
4 calls:
Slowest
14.0ms
(* (cbrt (sqrt (+ y x))) (cbrt (sqrt (+ y x))))
2.0ms
(cbrt (sqrt (+ y x)))
2.0ms
(cbrt (sqrt (+ y x)))
2.0ms
(cbrt (sqrt (+ y x)))

series498.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
137.0ms
(cbrt (sqrt (+ y x)))
129.0ms
(* (cbrt (sqrt (+ y x))) (cbrt (sqrt (+ y x))))
126.0ms
(cbrt (sqrt (+ y x)))
105.0ms
(cbrt (sqrt (+ y x)))

simplify2.3s

Counts
74 → 110
Calls
74 calls:
Slowest
280.0ms
(- (+ (pow x 1/3) (* 1/3 (* (pow (/ 1 (pow x 2)) 1/3) y))) (* 1/9 (* (pow (/ 1 (pow x 5)) 1/3) (pow y 2))))
267.0ms
(- (+ (* 1/6 (* (pow (/ 1 (pow x 5)) 1/6) y)) (pow x 1/6)) (* 5/72 (* (pow (/ 1 (pow x 11)) 1/6) (pow y 2))))
210.0ms
(- (+ (* 1/6 (* (pow (/ 1 (pow x 5)) 1/6) y)) (pow x 1/6)) (* 5/72 (* (pow (/ 1 (pow x 11)) 1/6) (pow y 2))))
193.0ms
(- (+ (* 1/6 (* (pow (/ 1 (pow x 5)) 1/6) y)) (pow x 1/6)) (* 5/72 (* (pow (/ 1 (pow x 11)) 1/6) (pow y 2))))
145.0ms
(* (cbrt (sqrt (- (* y y) (* x x)))) (cbrt (sqrt (+ y x))))

prune2.5s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.1b

regimes787.0ms

Accuracy

0% (0.2b remaining)

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

bsearch3.0ms

end0.0ms

sample16.4s

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