Average Error: 0.1 → 0.1
Time: 36.9s
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[(\left(\frac{a - \frac{1.0}{3.0}}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\right) \cdot rand + \left(a - \frac{1.0}{3.0}\right))_*\]

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.1

    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

  2. Initial simplification0.1

    \[\leadsto (\left(\frac{a - \frac{1.0}{3.0}}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot rand + \left(a - \frac{1.0}{3.0}\right))_*\]

  3. Final simplification0.1

    \[\leadsto (\left(\frac{a - \frac{1.0}{3.0}}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\right) \cdot rand + \left(a - \frac{1.0}{3.0}\right))_*\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))

Details

Time bar (total: 21.1s)Debug log

start523.0ms

Algorithm
intervals

setup208.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (- a (/ 1.0 3.0)) (sqrt (* 9 (- a (/ 1.0 3.0)))))
0.3b
(sqrt (* 9 (- a (/ 1.0 3.0))))
0.1b
(fma (/ (- a (/ 1.0 3.0)) (sqrt (* 9 (- a (/ 1.0 3.0))))) rand (- a (/ 1.0 3.0)))
0.0b
(* 9 (- a (/ 1.0 3.0)))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 80
Calls

4 calls. Slowest were:

4.0ms
(/ (- a (/ 1.0 3.0)) (sqrt (* 9 (- a (/ 1.0 3.0)))))
3.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))
1.0ms
(* 9 (- a (/ 1.0 3.0)))

series134.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

61.0ms
(fma (/ (- a (/ 1.0 3.0)) (sqrt (* 9 (- a (/ 1.0 3.0))))) rand (- a (/ 1.0 3.0)))
30.0ms
(/ (- a (/ 1.0 3.0)) (sqrt (* 9 (- a (/ 1.0 3.0)))))
25.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))
18.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify3.4s

Counts
61 → 92
Calls

61 calls. Slowest were:

346.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
311.0ms
(/ (* (* (- a (/ 1.0 3.0)) (- a (/ 1.0 3.0))) (- a (/ 1.0 3.0))) (* (* (sqrt (* 9 (- a (/ 1.0 3.0)))) (sqrt (* 9 (- a (/ 1.0 3.0))))) (sqrt (* 9 (- a (/ 1.0 3.0))))))
262.0ms
(sqrt (* 9 (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0)))))

prune1.3s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize26.0ms

Local error

Found 3 expressions with local error:

0.3b
(/ (sqrt (- a (/ 1.0 3.0))) 3)
0.1b
(fma (* (/ (sqrt (- a (/ 1.0 3.0))) 3) 1) rand (- a (/ 1.0 3.0)))
0.0b
(sqrt (- a (/ 1.0 3.0)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 46
Calls

3 calls. Slowest were:

6.0ms
(/ (sqrt (- a (/ 1.0 3.0))) 3)
2.0ms
(sqrt (- a (/ 1.0 3.0)))
0.0ms
(fma (* (/ (sqrt (- a (/ 1.0 3.0))) 3) 1) rand (- a (/ 1.0 3.0)))

series103.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

40.0ms
(/ (sqrt (- a (/ 1.0 3.0))) 3)
38.0ms
(fma (* (/ (sqrt (- a (/ 1.0 3.0))) 3) 1) rand (- a (/ 1.0 3.0)))
24.0ms
(sqrt (- a (/ 1.0 3.0)))

simplify1.6s

Counts
22 → 55
Calls

22 calls. Slowest were:

375.0ms
(sqrt (- (pow a 3) (pow (/ 1.0 3.0) 3)))
372.0ms
(sqrt (- (pow a 3) (pow (/ 1.0 3.0) 3)))
221.0ms
(sqrt (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))))

prune794.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

localize20.0ms

Local error

Found 3 expressions with local error:

0.4b
(* (sqrt (- a (/ 1.0 3.0))) (/ 1 3))
0.1b
(fma (* (* (sqrt (- a (/ 1.0 3.0))) (/ 1 3)) 1) rand (- a (/ 1.0 3.0)))
0.0b
(sqrt (- a (/ 1.0 3.0)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
3 → 55
Calls

3 calls. Slowest were:

4.0ms
(* (sqrt (- a (/ 1.0 3.0))) (/ 1 3))
1.0ms
(sqrt (- a (/ 1.0 3.0)))
0.0ms
(fma (* (* (sqrt (- a (/ 1.0 3.0))) (/ 1 3)) 1) rand (- a (/ 1.0 3.0)))

series85.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

35.0ms
(fma (* (* (sqrt (- a (/ 1.0 3.0))) (/ 1 3)) 1) rand (- a (/ 1.0 3.0)))
27.0ms
(sqrt (- a (/ 1.0 3.0)))
24.0ms
(* (sqrt (- a (/ 1.0 3.0))) (/ 1 3))

simplify2.3s

Counts
33 → 64
Calls

33 calls. Slowest were:

284.0ms
(* (sqrt (- (pow a 3) (pow (/ 1.0 3.0) 3))) 1)
276.0ms
(* (sqrt (- (pow a 3) (pow (/ 1.0 3.0) 3))) (/ 1 3))
275.0ms
(* (* (* (sqrt (- a (/ 1.0 3.0))) (sqrt (- a (/ 1.0 3.0)))) (sqrt (- a (/ 1.0 3.0)))) (* (* (/ 1 3) (/ 1 3)) (/ 1 3)))

prune794.0ms

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0b

localize22.0ms

Local error

Found 4 expressions with local error:

0.3b
(sqrt (* 9 (- a (/ 1.0 3.0))))
0.3b
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
0.2b
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
0.1b
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 81
Calls

4 calls. Slowest were:

7.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
7.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
4.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

series181.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

74.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
44.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
38.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
24.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify8.0s

Counts
50 → 93
Calls

50 calls. Slowest were:

794.0ms
(/ 1 (sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3)))))
736.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
472.0ms
(+ (* 1 1) (- (* (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand) (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)) (* 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))

prune1.4s

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0b

regimes156.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch4.0ms