Average Error: 0.1 → 0.1
Time: 46.7s
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(\frac{rand}{3 \cdot \sqrt{a - \frac{1.0}{3.0}}} + 1\right) \cdot \left(a - \frac{1.0}{3.0}\right)\]

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 sqrt-prod0.2

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

  4. Simplified0.2

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\color{blue}{3} \cdot \sqrt{a - \frac{1.0}{3.0}}} \cdot rand\right)\]
  5. Using strategy rm

  6. Applied associate-*l/0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{\frac{1 \cdot rand}{3 \cdot \sqrt{a - \frac{1.0}{3.0}}}}\right)\]

  7. Simplified0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{\color{blue}{rand}}{3 \cdot \sqrt{a - \frac{1.0}{3.0}}}\right)\]

  8. Final simplification0.1

    \[\leadsto \left(\frac{rand}{3 \cdot \sqrt{a - \frac{1.0}{3.0}}} + 1\right) \cdot \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2018360 
(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: 30.4s)Debug log

start829.0ms

Algorithm
intervals

setup361.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize21.0ms

Local error

Found 4 expressions with local error:

0.3b
(sqrt (* 9 (- a (/ 1.0 3.0))))
0.2b
(/ 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)))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 73
Calls

4 calls. Slowest were:

7.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
5.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
1.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))

series203.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

68.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
52.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
46.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
37.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify6.2s

Counts
50 → 85
Calls

50 calls. Slowest were:

371.0ms
(/ 1 (sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3)))))
365.0ms
(* (+ 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)))))
362.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

prune1.2s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

0.3b
(* 3 (sqrt (- a (/ 1.0 3.0))))
0.2b
(/ 1 (* 3 (sqrt (- a (/ 1.0 3.0)))))
0.2b
(* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)
0.1b
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
4 → 74
Calls

4 calls. Slowest were:

7.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)))
6.0ms
(* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)
2.0ms
(/ 1 (* 3 (sqrt (- a (/ 1.0 3.0)))))

series231.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

95.0ms
(* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)
49.0ms
(/ 1 (* 3 (sqrt (- a (/ 1.0 3.0)))))
44.0ms
(* 3 (sqrt (- a (/ 1.0 3.0))))
43.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)))

simplify6.8s

Counts
51 → 86
Calls

51 calls. Slowest were:

769.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (- (* 1 1) (* (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand) (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand))))
679.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)))
665.0ms
(* (+ a (/ 1.0 3.0)) (+ (* 1 1) (- (* (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand) (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)) (* 1 (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)))))

prune1.3s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

0.3b
(* 3 (sqrt (- a (/ 1.0 3.0))))
0.2b
(/ rand (* 3 (sqrt (- a (/ 1.0 3.0)))))
0.1b
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (* 3 (sqrt (- a (/ 1.0 3.0)))))))
0.0b
(+ 1 (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

5.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (* 3 (sqrt (- a (/ 1.0 3.0)))))))
2.0ms
(* 3 (sqrt (- a (/ 1.0 3.0))))
1.0ms
(+ 1 (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))))

series177.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

61.0ms
(/ rand (* 3 (sqrt (- a (/ 1.0 3.0)))))
43.0ms
(+ 1 (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))))
38.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (* 3 (sqrt (- a (/ 1.0 3.0)))))))
36.0ms
(* 3 (sqrt (- a (/ 1.0 3.0))))

simplify4.6s

Counts
45 → 78
Calls

45 calls. Slowest were:

385.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (/ rand (* 3 (sqrt (- a (/ 1.0 3.0)))))))
376.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (- (* 1 1) (* (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))) (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))))))
373.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (/ rand (* 3 (sqrt (- a (/ 1.0 3.0)))))))

prune1.2s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

0.3b
(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
(* (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
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)

series259.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

113.0ms
(* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)
62.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
53.0ms
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)
31.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify5.2s

Counts
49 → 79
Calls

49 calls. Slowest were:

405.0ms
(* (+ a (/ 1.0 3.0)) (- 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
361.0ms
(+ (pow -3.0 -1/2) (+ (* 1.5 (* a (sqrt -0.3333333333333333))) (+ (* 81/4 (/ (* (pow a 2) (sqrt -0.3333333333333333)) (pow -3.0 2))) (* 1.125 (* (pow a 2) (sqrt -0.3333333333333333))))))
314.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))

prune1.5s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

regimes265.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms