Average Error: 0.1 → 0.0
Time: 13.9s
Precision: 64
Internal Precision: 128
\[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\]
↓
\[(d1 \cdot \left(3 + d2\right) + \left(d3 \cdot d1\right))_*\]

Error

Bits error versus d1

Bits error versus d2

Bits error versus d3

Target

Original0.1
Target0.1
Herbie0.0
\[d1 \cdot \left(\left(3 + d2\right) + d3\right)\]

Derivation

  1. Initial program 0.1

    \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\]

  2. Simplified0.1

    \[\leadsto \color{blue}{(d1 \cdot \left(d2 + d3\right) + \left(d1 \cdot 3\right))_*}\]

  3. Taylor expanded around -inf 0.1

    \[\leadsto \color{blue}{d2 \cdot d1 + \left(d3 \cdot d1 + 3 \cdot d1\right)}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{(d1 \cdot \left(3 + d2\right) + \left(d3 \cdot d1\right))_*}\]

  5. Final simplification0.0

    \[\leadsto (d1 \cdot \left(3 + d2\right) + \left(d3 \cdot d1\right))_*\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (d1 d2 d3)
  :name "FastMath test3"

  :herbie-target
  (* d1 (+ (+ 3 d2) d3))

  (+ (+ (* d1 3) (* d1 d2)) (* d1 d3)))

Details

Time bar (total: 13.2s)Debug log

sample40.0ms

Algorithm
intervals

simplify89.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

89.0ms
(+ (+ (* d1 3) (* d1 d2)) (* d1 d3))

prune22.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize22.0ms

Local error

Found 1 expressions with local error:

0.0b
(fma d1 (+ d2 d3) (* d1 3))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(fma d1 (+ d2 d3) (* d1 3))

series34.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

34.0ms
(fma d1 (+ d2 d3) (* d1 3))

simplify174.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

68.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
60.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
45.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))

prune146.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 1 expressions with local error:

0.0b
(fma d1 (+ 3 d2) (* d3 d1))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(fma d1 (+ 3 d2) (* d3 d1))

series35.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

35.0ms
(fma d1 (+ 3 d2) (* d3 d1))

simplify204.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

78.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
71.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
54.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))

prune108.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
0.6b
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
0.6b
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
0.5b
(* (cbrt (fma d1 (+ 3 d2) (* d3 d1))) (cbrt (fma d1 (+ 3 d2) (* d3 d1))))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (fma d1 (+ 3 d2) (* d3 d1))) (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
1.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
1.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))

series512.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

148.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
137.0ms
(* (cbrt (fma d1 (+ 3 d2) (* d3 d1))) (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
126.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
100.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))

simplify3.9s

Counts
41 → 81
Calls

41 calls. Slowest were:

350.0ms
(+ (* 1/9 (* d2 (exp (* 1/3 (+ (log 3) (log d1)))))) (+ (* 1/9 (* d3 (exp (* 1/3 (+ (log 3) (log d1)))))) (exp (* 1/3 (+ (log 3) (log d1))))))
303.0ms
(+ (* 1/9 (* d2 (exp (* 1/3 (+ (log 3) (log d1)))))) (+ (* 1/9 (* d3 (exp (* 1/3 (+ (log 3) (log d1)))))) (exp (* 1/3 (+ (log 3) (log d1))))))
290.0ms
(+ (exp (* 1/3 (+ (log 9) (* 2 (log d1))))) (+ (* 2/9 (* (exp (* 1/3 (+ (log 9) (* 2 (log d1))))) d3)) (* 2/9 (* (exp (* 1/3 (+ (log 9) (* 2 (log d1))))) d2))))

prune1.0s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
0.6b
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
0.6b
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
0.6b
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

1.0ms
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
1.0ms
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
1.0ms
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))

series537.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

148.0ms
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
147.0ms
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
132.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
110.0ms
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))

simplify3.7s

Counts
33 → 73
Calls

33 calls. Slowest were:

575.0ms
(- (+ (/ (exp (* -1/3 (+ (log (/ 1 d1)) (log (/ 1 d2))))) d2) (exp (* -1/3 (+ (log (/ 1 d1)) (log (/ 1 d2)))))) (/ (exp (* -1/3 (+ (log (/ 1 d1)) (log (/ 1 d2))))) (pow d2 2)))
341.0ms
(+ (exp (* 1/9 (+ (log 3) (log d1)))) (+ (* 1/27 (* d2 (exp (* 1/9 (+ (log 3) (log d1)))))) (* 1/27 (* d3 (exp (* 1/9 (+ (log 3) (log d1))))))))
337.0ms
(+ (exp (* 1/9 (+ (log 3) (log d1)))) (+ (* 1/27 (* d2 (exp (* 1/9 (+ (log 3) (log d1)))))) (* 1/27 (* d3 (exp (* 1/9 (+ (log 3) (log d1))))))))

prune923.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

regimes130.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample1.5s

Algorithm
intervals