Average Error: 0.2 → 0.3
Time: 44.4s
Precision: 64
Internal Precision: 128
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\log \left(\sqrt[3]{x + y}\right) + \left((\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_* - t\right)\right) + \log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right)\]

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 0.2

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

  2. Initial simplification0.3

    \[\leadsto \log \left(y + x\right) + \left((\left(a - 0.5\right) \cdot \left(\log t\right) + \left(\log z\right))_* - t\right)\]
  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 2018365 +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: 24.9s)Debug log

start565.0ms

Algorithm
intervals

setup420.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize26.0ms

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)

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
3 → 55
Calls

3 calls. Slowest were:

16.0ms
(+ (log (+ y x)) (- (fma (- a 0.5) (log t) (log z)) t))
6.0ms
(- (fma (- a 0.5) (log t) (log z)) t)
0.0ms
(fma (- a 0.5) (log t) (log z))

series263.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

117.0ms
(+ (log (+ y x)) (- (fma (- a 0.5) (log t) (log z)) t))
97.0ms
(fma (- a 0.5) (log t) (log z))
48.0ms
(- (fma (- a 0.5) (log t) (log z)) t)

simplify3.2s

Counts
25 → 64
Calls

25 calls. Slowest were:

496.0ms
(- (+ (* 0.5 (log (/ -1 t))) (+ (* a (log -1)) (* 0.5 (log -1)))) (+ t (+ (* a (log (/ -1 t))) (log (/ -1 z)))))
494.0ms
(- (* 0.5 (log (/ 1 t))) (+ (log (/ 1 z)) (* a (log (/ 1 t)))))
482.0ms
(- (* 0.5 (log (/ 1 t))) (+ t (+ (log (/ 1 z)) (* a (log (/ 1 t))))))

prune963.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.1b

localize19.0ms

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
Counts
4 → 83
Calls

4 calls. Slowest were:

10.0ms
(* (cbrt (+ y x)) (cbrt (+ y x)))
1.0ms
(cbrt (+ y x))
1.0ms
(cbrt (+ y x))

series636.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

218.0ms
(* (cbrt (+ y x)) (cbrt (+ y x)))
149.0ms
(cbrt (+ y x))
149.0ms
(cbrt (+ y x))
119.0ms
(cbrt (+ y x))

simplify2.3s

Counts
59 → 95
Calls

59 calls. Slowest were:

287.0ms
(- (+ (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))))
278.0ms
(- (+ (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))))
272.0ms
(- (+ (* 2/3 (* (pow (/ 1 x) 1/3) y)) (pow x 2/3)) (* 1/9 (* (pow (/ 1 (pow x 4)) 1/3) (pow y 2))))

prune2.2s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.1b

localize23.0ms

Local error

Found 4 expressions with local error:

5.9b
(pow (+ y x) 2/3)
0.5b
(cbrt (+ y x))
0.4b
(fma (- a 0.5) (log t) (log z))
0.2b
(+ (log (cbrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

17.0ms
(+ (log (cbrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t))
2.0ms
(pow (+ y x) 2/3)
1.0ms
(cbrt (+ y x))

series638.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

224.0ms
(+ (log (cbrt (+ y x))) (- (fma (- a 0.5) (log t) (log z)) t))
181.0ms
(cbrt (+ y x))
148.0ms
(pow (+ y x) 2/3)
85.0ms
(fma (- a 0.5) (log t) (log z))

simplify3.1s

Counts
38 → 84
Calls

38 calls. Slowest were:

528.0ms
(- (+ (log (pow x 1/3)) (+ (* a (log t)) (log z))) (+ t (* 0.5 (log t))))
467.0ms
(- (* 0.5 (log (/ 1 t))) (+ (log (/ 1 z)) (* a (log (/ 1 t)))))
335.0ms
(- (+ (* 0.5 (log (/ -1 t))) (+ (* a (log -1)) (* 0.5 (log -1)))) (+ (* a (log (/ -1 t))) (log (/ -1 z))))

prune2.1s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.1b

localize16.0ms

Local error

Found 4 expressions with local error:

5.9b
(pow (+ y x) 2/3)
0.4b
(fma (- a 0.5) (log t) (log z))
0.1b
(fma 1/3 (log (+ y x)) (- (fma (- a 0.5) (log t) (log z)) t))
0.1b
(+ (log (pow (+ y x) 2/3)) (fma 1/3 (log (+ y x)) (- (fma (- a 0.5) (log t) (log z)) t)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

9.0ms
(+ (log (pow (+ y x) 2/3)) (fma 1/3 (log (+ y x)) (- (fma (- a 0.5) (log t) (log z)) t)))
2.0ms
(pow (+ y x) 2/3)
0.0ms
(fma 1/3 (log (+ y x)) (- (fma (- a 0.5) (log t) (log z)) t))

series666.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

255.0ms
(+ (log (pow (+ y x) 2/3)) (fma 1/3 (log (+ y x)) (- (fma (- a 0.5) (log t) (log z)) t)))
184.0ms
(fma 1/3 (log (+ y x)) (- (fma (- a 0.5) (log t) (log z)) t))
144.0ms
(pow (+ y x) 2/3)
82.0ms
(fma (- a 0.5) (log t) (log z))

simplify4.2s

Counts
28 → 72
Calls

28 calls. Slowest were:

789.0ms
(- (+ (log (pow x 2/3)) (+ (* a (log t)) (+ (log z) (* 1/3 (log x))))) (+ t (* 0.5 (log t))))
588.0ms
(- (* 0.5 (log (/ 1 t))) (+ (log (/ 1 z)) (* a (log (/ 1 t)))))
459.0ms
(- (+ (log (pow (/ 1 y) -2/3)) (* 0.5 (log (/ 1 t)))) (+ t (+ (log (/ 1 z)) (+ (* a (log (/ 1 t))) (* 1/3 (log (/ 1 y)))))))

prune1.6s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.1b

regimes1.8s

Accuracy

0% (0.2b remaining)

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

bsearch4.0ms