Average Error: 0.1 → 0.1
Time: 31.2s
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. 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 2019004 
(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: 30.5s)Debug log

sample326.0ms

Algorithm
intervals

simplify496.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

495.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize46.0ms

Local error

Found 4 expressions with local error:

0.6b
(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)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 73
Calls

4 calls. Slowest were:

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

series208.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

90.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
47.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
41.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
31.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify6.2s

Counts
50 → 85
Calls

50 calls. Slowest were:

556.0ms
(* (- a (/ 1.0 3.0)) (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))
421.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
399.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

prune811.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

0.6b
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)
0.2b
(* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)
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)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

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

series201.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

73.0ms
(* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)
59.0ms
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)
49.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
20.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify4.3s

Counts
49 → 79
Calls

49 calls. Slowest were:

397.0ms
(* (+ a (/ 1.0 3.0)) (+ (* 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)))))
330.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
313.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))

prune684.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))
0.2b
(* (pow (- a (/ 1.0 3.0)) -1/2) rand)
0.1b
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
0.0b
(pow (- a (/ 1.0 3.0)) -1/2)

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

7.0ms
(* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))
4.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
3.0ms
(* (pow (- a (/ 1.0 3.0)) -1/2) rand)

series259.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

70.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
63.0ms
(* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))
63.0ms
(pow (- a (/ 1.0 3.0)) -1/2)
62.0ms
(* (pow (- a (/ 1.0 3.0)) -1/2) rand)

simplify4.5s

Counts
49 → 81
Calls

49 calls. Slowest were:

404.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (- (* 1 1) (* (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand)) (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand)))))
374.0ms
(* (+ a (/ 1.0 3.0)) (- 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
346.0ms
(+ (* 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)))))

prune871.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (* (pow 9 -1/2) rand) (sqrt (- a (/ 1.0 3.0))))
0.3b
(* (pow 9 -1/2) rand)
0.0b
(+ (* 1 (- a (/ 1.0 3.0))) (* (* (pow 9 -1/2) rand) (sqrt (- a (/ 1.0 3.0)))))
0.0b
(sqrt (- a (/ 1.0 3.0)))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

10.0ms
(+ (* 1 (- a (/ 1.0 3.0))) (* (* (pow 9 -1/2) rand) (sqrt (- a (/ 1.0 3.0)))))
7.0ms
(* (* (pow 9 -1/2) rand) (sqrt (- a (/ 1.0 3.0))))
2.0ms
(* (pow 9 -1/2) rand)

series149.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

50.0ms
(+ (* 1 (- a (/ 1.0 3.0))) (* (* (pow 9 -1/2) rand) (sqrt (- a (/ 1.0 3.0)))))
43.0ms
(* (pow 9 -1/2) rand)
38.0ms
(* (* (pow 9 -1/2) rand) (sqrt (- a (/ 1.0 3.0))))
17.0ms
(sqrt (- a (/ 1.0 3.0)))

simplify3.0s

Counts
45 → 76
Calls

45 calls. Slowest were:

540.0ms
(* (* (* (* (pow 9 -1/2) rand) (* (pow 9 -1/2) rand)) (* (pow 9 -1/2) rand)) (* (* (sqrt (- a (/ 1.0 3.0))) (sqrt (- a (/ 1.0 3.0)))) (sqrt (- a (/ 1.0 3.0)))))
307.0ms
(sqrt (- (pow a 3) (pow (/ 1.0 3.0) 3)))
295.0ms
(- (+ (* 1/3 (* (sqrt (- 0.3333333333333333)) rand)) (* 1/6 (/ (* a rand) (sqrt (- 0.3333333333333333))))) (* 1/24 (/ (* (pow a 2) rand) (pow (sqrt (- 0.3333333333333333)) 3))))

prune991.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

regimes195.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms

end0.0ms

sample7.0s

Algorithm
intervals