Average Error: 0.1 → 0.1
Time: 39.9s
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + {\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\frac{-1}{2}} \cdot rand\right)\]

Error

Bits error versus a

Bits error versus rand

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied pow1/20.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\color{blue}{{\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\frac{1}{2}}}} \cdot rand\right)\]

  4. Applied pow-flip0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{{\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\left(-\frac{1}{2}\right)}} \cdot rand\right)\]

  5. Simplified0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + {\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\color{blue}{\frac{-1}{2}}} \cdot rand\right)\]
  6. Using strategy rm

  7. Applied *-commutative0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + {\color{blue}{\left(\left(a - \frac{1.0}{3.0}\right) \cdot 9\right)}}^{\frac{-1}{2}} \cdot rand\right)\]

  8. Final simplification0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + {\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\frac{-1}{2}} \cdot rand\right)\]

Reproduce

herbie shell --seed 2019008 +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: 39.1s)Debug log

sample391.0ms

Algorithm
intervals

simplify195.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

195.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize41.0ms

Local error

Found 4 expressions with local error:

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

rewrite22.0ms

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

series212.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

91.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
54.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
40.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
27.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify7.7s

Counts
50 → 93
Calls

50 calls. Slowest were:

549.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
485.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ (pow 1 3) (pow (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand) 3)))
445.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

prune936.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize38.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)
0.1b
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)
0.1b
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
0.0b
(* 9 (- a (/ 1.0 3.0)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 75
Calls

4 calls. Slowest were:

8.0ms
(* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)
8.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
3.0ms
(* 9 (- a (/ 1.0 3.0)))

series194.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

66.0ms
(* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)
65.0ms
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)
46.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
17.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify6.0s

Counts
49 → 87
Calls

49 calls. Slowest were:

554.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
502.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
450.0ms
(+ (* 1 1) (- (* (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand) (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)) (* 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand))))

prune963.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)
0.1b
(pow (* (- a (/ 1.0 3.0)) 9) -1/2)
0.1b
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)))
0.0b
(* (- a (/ 1.0 3.0)) 9)

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

6.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)))
5.0ms
(* (- a (/ 1.0 3.0)) 9)
5.0ms
(* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)

series206.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

89.0ms
(* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)
51.0ms
(pow (* (- a (/ 1.0 3.0)) 9) -1/2)
50.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)))
17.0ms
(* (- a (/ 1.0 3.0)) 9)

simplify6.2s

Counts
44 → 83
Calls

44 calls. Slowest were:

590.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)))
416.0ms
(+ (* 1 1) (- (* (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand) (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)) (* 1 (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand))))
403.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)))

prune1.0s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0b

localize28.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (pow 9 -1/2) rand)
0.3b
(* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand))
0.1b
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand))))
0.0b
(pow (- a (/ 1.0 3.0)) -1/2)

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 78
Calls

4 calls. Slowest were:

8.0ms
(* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand))
4.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand))))
2.0ms
(* (pow 9 -1/2) rand)

series248.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

80.0ms
(* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand))
67.0ms
(pow (- a (/ 1.0 3.0)) -1/2)
54.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand))))
47.0ms
(* (pow 9 -1/2) rand)

simplify6.3s

Counts
50 → 90
Calls

50 calls. Slowest were:

526.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand))))
523.0ms
(+ (* 0.16666666666666666 (/ (exp (* -1/2 (- (log -1) (log (/ -1 a))))) a)) (+ (exp (* -1/2 (- (log -1) (log (/ -1 a))))) (* 0.041666666666666664 (/ (exp (* -1/2 (- (log -1) (log (/ -1 a))))) (pow a 2)))))
491.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (- (* 1 1) (* (* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand)) (* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand)))))

prune880.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0b

regimes180.0ms

Accuracy

0% (0.1b remaining)

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

bsearch5.0ms

end0.0ms

sample7.3s

Algorithm
intervals