Average Error: 0.1 → 0.1
Time: 34.9s
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 
(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: 34.0s)Debug log

sample462.0ms

Algorithm
intervals

simplify422.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

422.0ms
(* (- 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

localize45.0ms

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)))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 73
Calls

4 calls. Slowest were:

8.0ms
(* (- 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
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))

series258.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

109.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
60.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
55.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
34.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify6.7s

Counts
50 → 85
Calls

50 calls. Slowest were:

711.0ms
(- (* 1/3 (* (sqrt -3.0) rand)) (+ (* 4.5 (/ (* (pow a 2) rand) (sqrt -3.0))) (+ (* 1.5 (/ (* a rand) (sqrt -3.0))) (* 3.375 (/ (* (pow a 2) rand) (pow (sqrt -3.0) 3))))))
453.0ms
(* (+ a (/ 1.0 3.0)) (+ (* 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)))))
448.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

prune787.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

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)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

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

series202.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

86.0ms
(* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)
50.0ms
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)
45.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
20.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify5.1s

Counts
49 → 79
Calls

49 calls. Slowest were:

425.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
414.0ms
(* (+ 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)))))
405.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))

prune632.0ms

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)

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

5.0ms
(* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)
4.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)))
2.0ms
(* (- a (/ 1.0 3.0)) 9)

series203.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

92.0ms
(* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)
50.0ms
(pow (* (- a (/ 1.0 3.0)) 9) -1/2)
44.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)))
18.0ms
(* (- a (/ 1.0 3.0)) 9)

simplify4.8s

Counts
44 → 75
Calls

44 calls. Slowest were:

441.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)))
385.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) 9)
351.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* (- a (/ 1.0 3.0)) 9) -1/2) rand)))

prune739.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0b

localize22.0ms

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)

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

10.0ms
(* (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)

series201.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

78.0ms
(* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand))
53.0ms
(pow (- a (/ 1.0 3.0)) -1/2)
37.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand))))
33.0ms
(* (pow 9 -1/2) rand)

simplify4.9s

Counts
50 → 82
Calls

50 calls. Slowest were:

438.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (- (* 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)))))
423.0ms
(* (- (* 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)))))
392.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (pow (- a (/ 1.0 3.0)) -1/2) (* (pow 9 -1/2) rand))))

prune1.0s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0b

regimes270.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms

end0.0ms

sample7.1s

Algorithm
intervals