Average Error: 0.0 → 0.7
Time: 19.5s
Precision: 64
Internal Precision: 128
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]
↓
\[(b \cdot a + \left(z \cdot t\right))_* + \left(\sqrt[3]{(i \cdot c + \left(y \cdot x\right))_*} \cdot \sqrt[3]{(i \cdot c + \left(y \cdot x\right))_*}\right) \cdot \sqrt[3]{(i \cdot c + \left(y \cdot x\right))_*}\]

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. Using strategy rm

  4. Applied add-cube-cbrt0.7

    \[\leadsto (b \cdot a + \left(z \cdot t\right))_* + \color{blue}{\left(\sqrt[3]{(i \cdot c + \left(x \cdot y\right))_*} \cdot \sqrt[3]{(i \cdot c + \left(x \cdot y\right))_*}\right) \cdot \sqrt[3]{(i \cdot c + \left(x \cdot y\right))_*}}\]

  5. Final simplification0.7

    \[\leadsto (b \cdot a + \left(z \cdot t\right))_* + \left(\sqrt[3]{(i \cdot c + \left(y \cdot x\right))_*} \cdot \sqrt[3]{(i \cdot c + \left(y \cdot x\right))_*}\right) \cdot \sqrt[3]{(i \cdot c + \left(y \cdot x\right))_*}\]

Reproduce

herbie shell --seed 2019010 +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: 18.7s)Debug log

sample128.0ms

Algorithm
intervals

simplify120.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
120.0ms
(+ (+ (+ (* x y) (* z t)) (* a b)) (* c i))

prune26.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize24.0ms

Local error

Found 3 expressions with local error:

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
6×*-un-lft-identity
5×add-log-exp
4×fma-udef
4×add-cube-cbrt
4×add-sqr-sqrt
3×log1p-expm1-u
3×add-exp-log
3×add-cbrt-cube
3×fma-def
3×pow1
3×expm1-log1p-u
1×distribute-lft-out
1×flip-+
1×flip3-+
1×sum-log
1×associate-+r+
1×associate-+l+
1×+-commutative
Counts
3 → 39
Calls
3 calls:
Slowest
4.0ms
(+ (fma b a (* z t)) (fma i c (* x y)))
0.0ms
(fma i c (* x y))
0.0ms
(fma b a (* z t))

series350.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
275.0ms
(+ (fma b a (* z t)) (fma i c (* x y)))
45.0ms
(fma i c (* x y))
29.0ms
(fma b a (* z t))

simplify351.0ms

Counts
15 → 48
Calls
15 calls:
Slowest
153.0ms
(* (exp (fma b a (* z t))) (exp (fma i c (* x y))))
73.0ms
(+ (fma b a (* z t)) (fma i c (* x y)))
30.0ms
(+ (* i c) (+ (* t z) (* x y)))
30.0ms
(+ (* i c) (+ (* t z) (* x y)))
10.0ms
(+ (* i c) (* x y))

prune716.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 4 expressions with local error:

1.5b
(cbrt (fma b a (* z t)))
1.5b
(cbrt (fma b a (* z t)))
1.5b
(cbrt (fma b a (* z t)))
0.5b
(* (cbrt (fma b a (* z t))) (cbrt (fma b a (* z t))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
15×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
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×expm1-log1p-u
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 69
Calls
4 calls:
Slowest
10.0ms
(* (cbrt (fma b a (* z t))) (cbrt (fma b a (* z t))))
1.0ms
(cbrt (fma b a (* z t)))
1.0ms
(cbrt (fma b a (* z t)))
1.0ms
(cbrt (fma b a (* z t)))

series558.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
163.0ms
(* (cbrt (fma b a (* z t))) (cbrt (fma b a (* z t))))
154.0ms
(cbrt (fma b a (* z t)))
127.0ms
(cbrt (fma b a (* z t)))
114.0ms
(cbrt (fma b a (* z t)))

simplify3.5s

Counts
41 → 81
Calls
41 calls:
Slowest
430.0ms
(exp (* -1/3 (+ (log (/ 1 b)) (log (/ 1 a)))))
402.0ms
(exp (* -1/3 (+ (log (/ 1 b)) (log (/ 1 a)))))
375.0ms
(exp (* -1/3 (+ (log (/ 1 b)) (log (/ 1 a)))))
373.0ms
(exp (* -1/3 (+ (* 2 (log (/ -1 a))) (* 2 (log (/ -1 b))))))
316.0ms
(+ (* 1/3 (* a (* (exp (* -1/3 (+ (* 2 (log t)) (* 2 (log z))))) b))) (exp (* 1/3 (+ (log z) (log t)))))

prune1.6s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

1.3b
(cbrt (fma i c (* x y)))
1.3b
(cbrt (fma i c (* x y)))
1.3b
(cbrt (fma i c (* x y)))
0.5b
(* (cbrt (fma i c (* x y))) (cbrt (fma i c (* x y))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
15×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
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×expm1-log1p-u
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 69
Calls
4 calls:
Slowest
10.0ms
(* (cbrt (fma i c (* x y))) (cbrt (fma i c (* x y))))
1.0ms
(cbrt (fma i c (* x y)))
1.0ms
(cbrt (fma i c (* x y)))
1.0ms
(cbrt (fma i c (* x y)))

series538.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
187.0ms
(* (cbrt (fma i c (* x y))) (cbrt (fma i c (* x y))))
123.0ms
(cbrt (fma i c (* x y)))
123.0ms
(cbrt (fma i c (* x y)))
105.0ms
(cbrt (fma i c (* x y)))

simplify3.0s

Counts
41 → 81
Calls
41 calls:
Slowest
346.0ms
(exp (* -1/3 (+ (* 2 (log (/ -1 i))) (* 2 (log (/ -1 c))))))
328.0ms
(exp (* -1/3 (+ (log (/ 1 c)) (log (/ 1 i)))))
318.0ms
(exp (* -1/3 (+ (log (/ 1 c)) (log (/ 1 i)))))
276.0ms
(exp (* -1/3 (+ (log (/ 1 c)) (log (/ 1 i)))))
255.0ms
(+ (exp (* 1/3 (+ (log y) (log x)))) (* 1/3 (* i (* c (exp (* -1/3 (+ (* 2 (log y)) (* 2 (log x)))))))))

prune1.3s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

1.5b
(cbrt (fma b a (* z t)))
1.5b
(cbrt (fma b a (* z t)))
1.5b
(cbrt (fma b a (* z t)))
1.5b
(cbrt (fma b a (* z t)))

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
12×cbrt-prod
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×pow1/3
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
Counts
4 → 52
Calls
4 calls:
Slowest
1.0ms
(cbrt (fma b a (* z t)))
1.0ms
(cbrt (fma b a (* z t)))
1.0ms
(cbrt (fma b a (* z t)))
1.0ms
(cbrt (fma b a (* z t)))

series506.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
152.0ms
(cbrt (fma b a (* z t)))
147.0ms
(cbrt (fma b a (* z t)))
109.0ms
(cbrt (fma b a (* z t)))
98.0ms
(cbrt (fma b a (* z t)))

simplify2.1s

Counts
24 → 64
Calls
24 calls:
Slowest
258.0ms
(exp (* -1/3 (+ (log (/ 1 b)) (log (/ 1 a)))))
245.0ms
(exp (* -1/3 (+ (log (/ 1 b)) (log (/ 1 a)))))
242.0ms
(exp (* -1/3 (+ (log (/ 1 b)) (log (/ 1 a)))))
242.0ms
(exp (* -1/3 (+ (log (/ 1 b)) (log (/ 1 a)))))
165.0ms
(+ (* 1/3 (* a (* (exp (* -1/3 (+ (* 2 (log t)) (* 2 (log z))))) b))) (exp (* 1/3 (+ (log z) (log t)))))

prune891.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

regimes683.0ms

Accuracy

0% (0.6b remaining)

Error of 0.7b against oracle of 0.0b and baseline of 0.7b

bsearch2.0ms

end0.0ms

sample2.2s

Algorithm
intervals