Average Error: 0.1 → 0.0
Time: 12.8s
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 2019004 +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: 12.2s)Debug log

sample38.0ms

Algorithm
intervals

simplify49.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

49.0ms
(+ (+ (* d1 3) (* d1 d2)) (* d1 d3))

prune16.0ms

Pruning

1 alts after pruning (1 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))

series32.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

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

simplify198.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

82.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
59.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
56.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))

prune164.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize17.0ms

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))

simplify230.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

79.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
76.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
75.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))

prune160.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

9.0ms
(* (cbrt (fma d1 (+ d2 d3) (* d1 3))) (cbrt (fma d1 (+ d2 d3) (* d1 3))))
1.0ms
(cbrt (fma d1 (+ d2 d3) (* d1 3)))
1.0ms
(cbrt (fma d1 (+ d2 d3) (* d1 3)))

series526.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

150.0ms
(* (cbrt (fma d1 (+ d2 d3) (* d1 3))) (cbrt (fma d1 (+ d2 d3) (* d1 3))))
134.0ms
(cbrt (fma d1 (+ d2 d3) (* d1 3)))
131.0ms
(cbrt (fma d1 (+ d2 d3) (* d1 3)))
112.0ms
(cbrt (fma d1 (+ d2 d3) (* d1 3)))

simplify3.7s

Counts
41 → 81
Calls

41 calls. Slowest were:

318.0ms
(+ (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))))
293.0ms
(+ (* 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))))))
262.0ms
(+ (* 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))))))

prune857.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (fma d1 (+ d2 d3) (* d1 3)))
0.6b
(cbrt (fma d1 (+ d2 d3) (* d1 3)))
0.3b
(cbrt (sqrt (fma d1 (+ d2 d3) (* d1 3))))
0.3b
(cbrt (sqrt (fma d1 (+ d2 d3) (* d1 3))))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 58
Calls

4 calls. Slowest were:

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

series445.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

134.0ms
(cbrt (sqrt (fma d1 (+ d2 d3) (* d1 3))))
110.0ms
(cbrt (sqrt (fma d1 (+ d2 d3) (* d1 3))))
102.0ms
(cbrt (fma d1 (+ d2 d3) (* d1 3)))
99.0ms
(cbrt (fma d1 (+ d2 d3) (* d1 3)))

simplify3.0s

Counts
30 → 70
Calls

30 calls. Slowest were:

340.0ms
(+ (exp (* 1/6 (+ (log 3) (log d1)))) (+ (* 1/18 (* d2 (exp (* 1/6 (+ (log 3) (log d1)))))) (* 1/18 (* d3 (exp (* 1/6 (+ (log 3) (log d1))))))))
317.0ms
(+ (exp (* 1/6 (+ (log 3) (log d1)))) (+ (* 1/18 (* d2 (exp (* 1/6 (+ (log 3) (log d1)))))) (* 1/18 (* d3 (exp (* 1/6 (+ (log 3) (log d1))))))))
241.0ms
(+ (* 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))))))

prune856.0ms

Pruning

4 alts after pruning (0 fresh and 4 done)

Merged error: 0.0b

regimes165.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample1.6s

Algorithm
intervals