Average Error: 0.1 → 0.1
Time: 33.6s
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 +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: 33.0s)Debug log

sample297.0ms

Algorithm
intervals

simplify170.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

170.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize22.0ms

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)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 81
Calls

4 calls. Slowest were:

4.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
4.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
2.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

series179.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

64.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
62.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
32.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
21.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify7.3s

Counts
50 → 93
Calls

50 calls. Slowest were:

698.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
516.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
482.0ms
(+ (* 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))))

prune1.0s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize32.0ms

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)))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 75
Calls

4 calls. Slowest were:

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

series224.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

73.0ms
(* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)
64.0ms
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)
58.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
29.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify6.3s

Counts
49 → 87
Calls

49 calls. Slowest were:

1.1s
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
508.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
497.0ms
(+ (* 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))))

prune859.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize35.0ms

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)

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 77
Calls

4 calls. Slowest were:

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

series271.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

82.0ms
(* (pow (- a (/ 1.0 3.0)) -1/2) rand)
69.0ms
(* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))
62.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
57.0ms
(pow (- a (/ 1.0 3.0)) -1/2)

simplify7.0s

Counts
49 → 89
Calls

49 calls. Slowest were:

694.0ms
(* (+ a (/ 1.0 3.0)) (- 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
626.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
561.0ms
(* (- (* 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)))))

prune1.0s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize35.0ms

Local error

Found 4 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

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

series169.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

67.0ms
(pow (- a (/ 1.0 3.0)) 1/2)
55.0ms
(fma (* (pow 9 -1/2) rand) (pow (- a (/ 1.0 3.0)) 1/2) (- a (/ 1.0 3.0)))
27.0ms
(* (pow 9 -1/2) rand)
19.0ms
(- a (/ 1.0 3.0))

simplify674.0ms

Counts
19 → 64
Calls

19 calls. Slowest were:

257.0ms
(- (+ (pow (- 0.3333333333333333) 1/2) (* 1.125 (* (pow a 2) (sqrt -0.3333333333333333)))) (+ (* 1.5 (* a (sqrt -0.3333333333333333))) (* 1/4 (/ (* (pow a 2) (sqrt -0.3333333333333333)) (pow -0.3333333333333333 2)))))
202.0ms
(- (exp (* 1/2 (- (log -1) (log (/ -1 a))))) (+ (* 0.013888888888888888 (/ (exp (* 1/2 (- (log -1) (log (/ -1 a))))) (pow a 2))) (* 0.16666666666666666 (/ (exp (* 1/2 (- (log -1) (log (/ -1 a))))) a))))
97.0ms
(- (pow (/ 1 a) -1/2) (+ (* 0.013888888888888888 (sqrt (/ 1 (pow a 3)))) (* 0.16666666666666666 (sqrt (/ 1 a)))))

prune759.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

regimes163.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms

end0.0ms

sample6.4s

Algorithm
intervals