Average Error: 0.0 → 0.0
Time: 19.2s
Precision: 64
Internal Precision: 128
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]
↓
\[(b \cdot a + \left(z \cdot t\right))_* + (i \cdot c + \left(y \cdot x\right))_*\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

  1. Initial program 0.0

    \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]

  2. Simplified0.0

    \[\leadsto \color{blue}{(b \cdot a + \left(z \cdot t\right))_* + (i \cdot c + \left(x \cdot y\right))_*}\]

  3. Final simplification0.0

    \[\leadsto (b \cdot a + \left(z \cdot t\right))_* + (i \cdot c + \left(y \cdot x\right))_*\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3"
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))

Details

Time bar (total: 18.5s)Debug log

sample114.0ms

Algorithm
intervals
Results
472×(pre true 80)
472×(body real 80)

simplify128.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
128.0ms
(+ (+ (+ (* x y) (* z t)) (* a b)) (* c i))

prune30.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize37.0ms

Local error

Found 2 expressions with local error:

0.0b
(+ (fma b a (* z t)) (fma i c (* x y)))
0.0b
(fma b a (* z t))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
5×*-un-lft-identity
4×add-log-exp
3×fma-udef
3×add-cube-cbrt
3×fma-def
3×add-sqr-sqrt
2×log1p-expm1-u
2×add-exp-log
2×add-cbrt-cube
2×pow1
2×expm1-log1p-u
1×distribute-lft-out
1×flip-+
1×flip3-+
1×sum-log
1×associate-+r+
1×associate-+l+
1×+-commutative
Counts
2 → 29
Calls
2 calls:
Slowest
4.0ms
(+ (fma b a (* z t)) (fma i c (* x y)))
0.0ms
(fma b a (* z t))

series287.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
259.0ms
(+ (fma b a (* z t)) (fma i c (* x y)))
28.0ms
(fma b a (* z t))

simplify287.0ms

Counts
12 → 35
Calls
12 calls:
Slowest
108.0ms
(* (exp (fma b a (* z t))) (exp (fma i c (* x y))))
70.0ms
(+ (fma b a (* z t)) (fma i c (* x y)))
28.0ms
(+ (* i c) (+ (* t z) (* x y)))
26.0ms
(+ (* i c) (+ (* t z) (* x y)))
15.0ms
(* (cbrt (fma b a (* z t))) (cbrt (fma b a (* z t))))

prune586.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (+ (fma b a (* z t)) (fma i c (* x y))))
0.6b
(cbrt (+ (fma b a (* z t)) (fma i c (* x y))))
0.6b
(cbrt (+ (fma b a (* z t)) (fma i c (* x y))))
0.5b
(* (cbrt (+ (fma b a (* z t)) (fma i c (* x y)))) (cbrt (+ (fma b a (* z t)) (fma i c (* x y)))))

rewrite41.0ms

Algorithm
rewrite-expression-head
Rules
21×*-un-lft-identity
20×cbrt-prod
18×cbrt-div
11×add-cube-cbrt
11×add-sqr-sqrt
9×flip-+
9×flip3-+
9×pow1
7×associate-*l*
7×associate-*r*
7×pow1/3
6×add-exp-log
5×distribute-lft-out
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 → 88
Calls
4 calls:
Slowest
25.0ms
(* (cbrt (+ (fma b a (* z t)) (fma i c (* x y)))) (cbrt (+ (fma b a (* z t)) (fma i c (* x y)))))
5.0ms
(cbrt (+ (fma b a (* z t)) (fma i c (* x y))))
4.0ms
(cbrt (+ (fma b a (* z t)) (fma i c (* x y))))
4.0ms
(cbrt (+ (fma b a (* z t)) (fma i c (* x y))))

series647.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
214.0ms
(* (cbrt (+ (fma b a (* z t)) (fma i c (* x y)))) (cbrt (+ (fma b a (* z t)) (fma i c (* x y)))))
156.0ms
(cbrt (+ (fma b a (* z t)) (fma i c (* x y))))
149.0ms
(cbrt (+ (fma b a (* z t)) (fma i c (* x y))))
128.0ms
(cbrt (+ (fma b a (* z t)) (fma i c (* x y))))

simplify9.4s

Counts
64 → 100
Calls
64 calls:
Slowest
511.0ms
(* (+ (fma b a (* z t)) (fma i c (* x y))) (+ (fma b a (* z t)) (fma i c (* x y))))
489.0ms
(* (+ (fma b a (* z t)) (fma i c (* x y))) (+ (fma b a (* z t)) (fma i c (* x y))))
410.0ms
(exp (* -1/3 (+ (* 2 (log (/ -1 a))) (* 2 (log (/ -1 b))))))
407.0ms
(* (cbrt (- (fma b a (* z t)) (fma i c (* x y)))) (cbrt (+ (* (fma b a (* z t)) (fma b a (* z t))) (- (* (fma i c (* x y)) (fma i c (* x y))) (* (fma b a (* z t)) (fma i c (* x y)))))))
396.0ms
(* (cbrt (+ (* (fma b a (* z t)) (fma b a (* z t))) (- (* (fma i c (* x y)) (fma i c (* x y))) (* (fma b a (* z t)) (fma i c (* x y)))))) (cbrt (- (fma b a (* z t)) (fma i c (* x y)))))

prune1.7s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 2 expressions with local error:

0.0b
(fma i c (* t z))
0.0b
(fma x y (fma i c (* t z)))

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
2×add-log-exp
2×log1p-expm1-u
2×fma-udef
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×*-un-lft-identity
2×pow1
2×expm1-log1p-u
2×add-sqr-sqrt
Counts
2 → 20
Calls
2 calls:
Slowest
0.0ms
(fma i c (* t z))
0.0ms
(fma x y (fma i c (* t z)))

series130.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
78.0ms
(fma x y (fma i c (* t z)))
52.0ms
(fma i c (* t z))

simplify48.0ms

Counts
6 → 26
Calls
6 calls:
Slowest
21.0ms
(+ (* i c) (+ (* t z) (* x y)))
15.0ms
(+ (* i c) (+ (* t z) (* x y)))
5.0ms
(+ (* t z) (* i c))
5.0ms
(+ (* t z) (* i c))
0.0ms
0

prune380.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.0b

regimes795.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample3.8s

Algorithm
intervals
Results
13689×(pre true 80)
13689×(body real 80)