Average Error: 0.1 → 0.0
Time: 14.5s
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. Initial simplification0.1

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

  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 2018362 +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: 11.7s)Debug log

start40.0ms

Algorithm
intervals

setup66.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize21.0ms

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))

series32.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

32.0ms
(fma d1 (+ d2 d3) (* d1 3))

simplify168.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

60.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
57.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
51.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))

prune116.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 1 expressions with local error:

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

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

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

series33.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

33.0ms
(fma d1 (+ 3 d2) (* d3 d1))

simplify172.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

61.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
61.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
50.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))

prune120.0ms

Pruning

4 alts after pruning (2 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)))
0.6b
(* (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:

6.0ms
(* (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)))

series463.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

142.0ms
(* (cbrt (fma d1 (+ 3 d2) (* d3 d1))) (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
114.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
112.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
94.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))

simplify4.1s

Counts
41 → 81
Calls

41 calls. Slowest were:

419.0ms
(+ (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))))
326.0ms
(+ (* 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))))))
306.0ms
(+ (* 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))))))

prune1.2s

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0.0b

localize31.0ms

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

1.0ms
(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)))

series470.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

142.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
114.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
113.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
102.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))

simplify3.4s

Counts
24 → 64
Calls

24 calls. Slowest were:

587.0ms
(+ (* 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))))))
343.0ms
(+ (* 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))))))
282.0ms
(+ (* 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))))))

prune1.1s

Pruning

6 alts after pruning (2 fresh and 4 done)

Merged error: 0.0b

regimes234.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms