Average Error: 0.0 → 0.0
Time: 52.8s
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 2019022 +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: 51.9s)Debug log

sample55.0ms

Algorithm
intervals
Results
37.0ms423×body80valid

simplify127.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
127.0ms
(+ (+ (+ (* x y) (* z t)) (* a b)) (* c i))

prune27.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize35.0ms

Local error

Found 1 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
4×*-un-lft-identity
3×add-log-exp
3×fma-def
2×fma-udef
2×add-cube-cbrt
2×add-sqr-sqrt
1×distribute-lft-out
1×log1p-expm1-u
1×flip-+
1×add-exp-log
1×add-cbrt-cube
1×flip3-+
1×sum-log
1×associate-+r+
1×associate-+l+
1×+-commutative
1×pow1
1×expm1-log1p-u
Counts
1 → 19
Calls
1 calls:
Slowest
9.0ms
(+ (fma b a (* z t)) (fma i c (* x y)))

series247.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
247.0ms
(+ (fma b a (* z t)) (fma i c (* x y)))

simplify245.0ms

Counts
9 → 22
Calls
9 calls:
Slowest
125.0ms
(* (exp (fma b a (* z t))) (exp (fma i c (* x y))))
35.0ms
(+ (fma b a (* z t)) (fma i c (* x y)))
27.0ms
(+ (* i c) (+ (* t z) (* x y)))
27.0ms
(+ (* i c) (+ (* t z) (* x y)))
13.0ms
(* (cbrt (fma b a (* z t))) (cbrt (fma b a (* z t))))

prune334.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize13.0ms

Local error

Found 2 expressions with local error:

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

series101.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
58.0ms
(fma x y (fma i c (* t z)))
43.0ms
(fma i c (* t z))

simplify78.0ms

Counts
6 → 26
Calls
6 calls:
Slowest
30.0ms
(+ (* i c) (+ (* t z) (* x y)))
27.0ms
(+ (* i c) (+ (* t z) (* x y)))
9.0ms
(+ (* t z) (* i c))
9.0ms
(+ (* t z) (* i c))
1.0ms
0

prune334.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize6.0ms

Local error

Found 4 expressions with local error:

0.3b
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))
0.3b
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))
0.3b
(* (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))))
0.0b
(+ (fma b a (* z t)) (fma i c (* x y)))

rewrite30.0ms

Algorithm
rewrite-expression-head
Rules
21×*-un-lft-identity
17×add-sqr-sqrt
16×sqrt-prod
16×sqrt-div
11×add-cube-cbrt
9×flip-+
9×flip3-+
9×pow1
7×associate-*l*
7×associate-*r*
6×add-log-exp
6×add-exp-log
6×pow1/2
6×add-cbrt-cube
5×distribute-lft-out
4×log1p-expm1-u
4×frac-times
4×expm1-log1p-u
3×fma-def
2×hypot-def
2×fma-udef
2×pow-prod-up
2×associate-*r/
2×associate-*l/
2×pow-prod-down
2×rem-sqrt-square
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×sqrt-unprod
1×pow-plus
1×rem-square-sqrt
1×sum-log
1×associate-+r+
1×associate-+l+
1×+-commutative
1×pow2
Counts
4 → 97
Calls
4 calls:
Slowest
14.0ms
(* (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))))
6.0ms
(+ (fma b a (* z t)) (fma i c (* x y)))
4.0ms
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))
4.0ms
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))

series10.3s

Counts
4 → 12
Calls
4 calls:
Slowest
4.9s
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))
4.9s
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))
241.0ms
(+ (fma b a (* z t)) (fma i c (* x y)))
237.0ms
(* (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))))

simplify7.3s

Counts
71 → 109
Calls
71 calls:
Slowest
550.0ms
(* (* (* (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))) (* (* (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))))
527.0ms
(* (+ (fma b a (* z t)) (fma i c (* x y))) (+ (fma b a (* z t)) (fma i c (* x y))))
466.0ms
(* (+ (fma b a (* z t)) (fma i c (* x y))) (+ (fma b a (* z t)) (fma i c (* x y))))
359.0ms
(sqrt (+ (* (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))))))
339.0ms
(* (sqrt (+ (pow (fma b a (* z t)) 3) (pow (fma i c (* x y)) 3))) (sqrt (- (* (fma b a (* z t)) (fma b a (* z t))) (* (fma i c (* x y)) (fma i c (* x y))))))

prune1.9s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

localize19.0ms

Local error

Found 4 expressions with local error:

11.3b
(cbrt (* (* (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))))
0.3b
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))
0.3b
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))
0.3b
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))

rewrite75.0ms

Algorithm
rewrite-expression-head
Rules
60×sqrt-div
30×flip-+
30×flip3-+
28×frac-times
26×cbrt-div
16×add-sqr-sqrt
14×associate-*l/
13×*-un-lft-identity
12×sqrt-prod
8×associate-*r/
7×add-cube-cbrt
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×distribute-lft-out
3×hypot-def
3×pow1/2
3×rem-sqrt-square
1×rem-cbrt-cube
1×pow3
1×pow1/3
1×cbrt-prod
Counts
4 → 92
Calls
4 calls:
Slowest
50.0ms
(cbrt (* (* (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))))
7.0ms
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))
7.0ms
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))
7.0ms
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))

series17.9s

Counts
4 → 12
Calls
4 calls:
Slowest
5.1s
(cbrt (* (* (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))))
4.3s
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))
4.3s
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))
4.2s
(sqrt (+ (fma b a (* z t)) (fma i c (* x y))))

simplify7.7s

Counts
64 → 104
Calls
64 calls:
Slowest
303.0ms
(cbrt (* (* (sqrt (- (* (fma b a (* z t)) (fma b a (* z t))) (* (fma i c (* x y)) (fma i c (* x y))))) (sqrt (+ (pow (fma b a (* z t)) 3) (pow (fma i c (* x y)) 3)))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))))
302.0ms
(cbrt (* (* (sqrt (+ (pow (fma b a (* z t)) 3) (pow (fma i c (* x y)) 3))) (sqrt (- (* (fma b a (* z t)) (fma b a (* z t))) (* (fma i c (* x y)) (fma i c (* x y)))))) (sqrt (+ (fma b a (* z t)) (fma i c (* x y))))))
302.0ms
(cbrt (* (* (sqrt (- (* (fma b a (* z t)) (fma b a (* z t))) (* (fma i c (* x y)) (fma i c (* x y))))) (sqrt (+ (pow (fma b a (* z t)) 3) (pow (fma i c (* x y)) 3)))) (sqrt (+ (pow (fma b a (* z t)) 3) (pow (fma i c (* x y)) 3)))))
297.0ms
(cbrt (* (* (sqrt (+ (fma b a (* z t)) (fma i c (* x y)))) (sqrt (+ (pow (fma b a (* z t)) 3) (pow (fma i c (* x y)) 3)))) (sqrt (- (* (fma b a (* z t)) (fma b a (* z t))) (* (fma i c (* x y)) (fma i c (* x y)))))))
284.0ms
(cbrt (* (* (sqrt (+ (pow (fma b a (* z t)) 3) (pow (fma i c (* x y)) 3))) (sqrt (- (* (fma b a (* z t)) (fma b a (* z t))) (* (fma i c (* x y)) (fma i c (* x y)))))) (sqrt (- (* (fma b a (* z t)) (fma b a (* z t))) (* (fma i c (* x y)) (fma i c (* x y)))))))

prune1.8s

Pruning

4 alts after pruning (0 fresh and 4 done)

Merged error: 0b

regimes716.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample2.6s

Algorithm
intervals
Results
2.0s13866×body80valid