Average Error: 0.3 → 0.3
Time: 47.2s
Precision: 64
Internal Precision: 128
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[(\left(\log t\right) \cdot \left(a - 0.5\right) + \left(\log \left(y + x\right)\right))_* - \left(t - \log z\right)\]

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. Initial simplification0.3

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2018360 +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: 28.0s)Debug log

start654.0ms

Algorithm
intervals

setup437.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize62.0ms

Local error

Found 4 expressions with local error:

0.6b
(+ (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)

rewrite240.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

169.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
51.0ms
(- (+ (log (+ x y)) (log z)) t)
9.0ms
(+ (log (+ x y)) (log z))

series541.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

223.0ms
(+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
170.0ms
(- (+ (log (+ x y)) (log z)) t)
91.0ms
(+ (log (+ x y)) (log z))
56.0ms
(* (- a 0.5) (log t))

simplify3.9s

Counts
52 → 102
Calls

52 calls. Slowest were:

445.0ms
(- (+ (* a (log t)) (+ (log z) (log y))) (+ t (* 0.5 (log t))))
403.0ms
(- (+ (* 0.5 (log (/ -1 t))) (* a (log -1))) (+ (* a (log (/ -1 t))) (* 0.5 (log -1))))
306.0ms
(* (exp (- (+ (log (+ x y)) (log z)) t)) (exp (* (- a 0.5) (log t))))

prune1.7s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 2 expressions with local error:

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 42
Calls

2 calls. Slowest were:

4.0ms
(- (fma (log t) (- a 0.5) (log (+ y x))) (- t (log z)))
0.0ms
(fma (log t) (- a 0.5) (log (+ y x)))

series222.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

124.0ms
(fma (log t) (- a 0.5) (log (+ y x)))
98.0ms
(- (fma (log t) (- a 0.5) (log (+ y x))) (- t (log z)))

simplify5.4s

Counts
34 → 48
Calls

34 calls. Slowest were:

354.0ms
(- (+ (* a (log t)) (+ (log z) (log x))) (+ t (* 0.5 (log t))))
326.0ms
(fma 1 (fma (log t) (- a 0.5) (log (+ y x))) (- (* (- t (log z)) 1)))
325.0ms
(- (+ (* 0.5 (log (/ -1 t))) (+ (* a (log -1)) (* 0.5 (log -1)))) (+ (* a (log (/ -1 t))) (log (/ -1 y))))

prune760.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

localize30.0ms

Local error

Found 4 expressions with local error:

0.5b
(+ (log (sqrt (+ x y))) (log z))
0.3b
(+ (log (sqrt (+ x y))) (+ (log (sqrt (+ x y))) (log z)))
0.2b
(* (- a 0.5) (log t))
0.1b
(+ (- (+ (log (sqrt (+ x y))) (+ (log (sqrt (+ x y))) (log z))) t) (* (- a 0.5) (log t)))

rewrite445.0ms

Algorithm
rewrite-expression-head
Counts
4 → 109
Calls

4 calls. Slowest were:

422.0ms
(+ (- (+ (log (sqrt (+ x y))) (+ (log (sqrt (+ x y))) (log z))) t) (* (- a 0.5) (log t)))
13.0ms
(+ (log (sqrt (+ x y))) (+ (log (sqrt (+ x y))) (log z)))
5.0ms
(+ (log (sqrt (+ x y))) (log z))

series472.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

219.0ms
(+ (- (+ (log (sqrt (+ x y))) (+ (log (sqrt (+ x y))) (log z))) t) (* (- a 0.5) (log t)))
94.0ms
(+ (log (sqrt (+ x y))) (+ (log (sqrt (+ x y))) (log z)))
86.0ms
(+ (log (sqrt (+ x y))) (log z))
72.0ms
(* (- a 0.5) (log t))

simplify4.2s

Counts
71 → 121
Calls

71 calls. Slowest were:

773.0ms
(* (+ (+ (log (sqrt (+ x y))) (+ (log (sqrt (+ x y))) (log z))) t) (+ a 0.5))
357.0ms
(- (+ (* 0.5 (log (/ -1 t))) (* a (log -1))) (+ (* a (log (/ -1 t))) (* 0.5 (log -1))))
347.0ms
(+ (* (- (pow (+ (log (sqrt (+ x y))) (+ (log (sqrt (+ x y))) (log z))) 3) (pow t 3)) (+ (* a a) (+ (* 0.5 0.5) (* a 0.5)))) (* (+ (* (+ (log (sqrt (+ x y))) (+ (log (sqrt (+ x y))) (log z))) (+ (log (sqrt (+ x y))) (+ (log (sqrt (+ x y))) (log z)))) (+ (* t t) (* (+ (log (sqrt (+ x y))) (+ (log (sqrt (+ x y))) (log z))) t))) (* (- (pow a 3) (pow 0.5 3)) (log t))))

prune2.4s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 4 expressions with local error:

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

rewrite23.0ms

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)))

series420.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

122.0ms
(cbrt (* (- a 0.5) (log t)))
108.0ms
(cbrt (* (- a 0.5) (log t)))
102.0ms
(+ (log (+ x y)) (log z))
89.0ms
(cbrt (* (- a 0.5) (log t)))

simplify3.1s

Counts
31 → 75
Calls

31 calls. Slowest were:

261.0ms
(- (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)))))
239.0ms
(- (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)))))
220.0ms
(cbrt (* (- (* a a) (* 0.5 0.5)) (log t)))

prune1.4s

Pruning

10 alts after pruning (9 fresh and 1 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

bsearch1.0ms