Average Error: 0.1 → 0.0
Time: 13.1s
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 2019008 +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: 12.5s)Debug log

sample65.0ms

Algorithm
intervals

simplify59.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune10.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize14.0ms

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))

series33.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

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

simplify203.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

79.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
62.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
60.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))

prune103.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 1 expressions with local error:

0.1b
(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))

simplify175.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

82.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
51.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))
40.0ms
(+ (* d2 d1) (+ (* d3 d1) (* 3 d1)))

prune120.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

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

rewrite7.0ms

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)))

series494.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

162.0ms
(* (cbrt (fma d1 (+ 3 d2) (* d3 d1))) (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
124.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
118.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
90.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))

simplify4.1s

Counts
41 → 81
Calls

41 calls. Slowest were:

378.0ms
(+ (* 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))))))
377.0ms
(+ (* 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))))))
307.0ms
(+ (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))))

prune834.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

localize22.0ms

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

rewrite3.0ms

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))))

series435.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

120.0ms
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
118.0ms
(cbrt (fma d1 (+ 3 d2) (* d3 d1)))
108.0ms
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))
89.0ms
(cbrt (cbrt (fma d1 (+ 3 d2) (* d3 d1))))

simplify3.2s

Counts
33 → 73
Calls

33 calls. Slowest were:

359.0ms
(+ (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))))))))
317.0ms
(+ (* 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
(+ (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))))))))

prune882.0ms

Pruning

4 alts after pruning (0 fresh and 4 done)

Merged error: 0.0b

regimes119.0ms

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