Average Error: 0.2 → 0.1
Time: 36.8s
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{\frac{rand}{3}}{\sqrt{a - \frac{1.0}{3.0}}}\right)\]

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 0.2

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

  3. Applied associate-*l/0.1

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

  4. Simplified0.1

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

  6. Applied sqrt-prod0.1

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

  7. Applied associate-/r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019007 +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: 36.2s)Debug log

sample374.0ms

Algorithm
intervals

simplify168.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

168.0ms
(* (- 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

localize31.0ms

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))))

series209.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

94.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
53.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
39.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
22.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify7.3s

Counts
50 → 93
Calls

50 calls. Slowest were:

719.0ms
(* (+ a (/ 1.0 3.0)) (- 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
614.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
464.0ms
(+ (* 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))))

prune948.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 4 expressions with local error:

0.6b
(sqrt (* 9 (- a (/ 1.0 3.0))))
0.3b
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
0.1b
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
0.0b
(* 9 (- a (/ 1.0 3.0)))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

11.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
4.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))
3.0ms
(* 9 (- a (/ 1.0 3.0)))

series161.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

54.0ms
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
40.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
37.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))
29.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify5.5s

Counts
52 → 91
Calls

52 calls. Slowest were:

615.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
446.0ms
(* (+ a (/ 1.0 3.0)) (+ (* 1 1) (- (* (/ rand (sqrt (* 9 (- a (/ 1.0 3.0))))) (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))) (* 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))))
415.0ms
(- (* 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))))))

prune855.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 4 expressions with local error:

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 109
Calls

4 calls. Slowest were:

12.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ (/ rand (sqrt 9)) (sqrt (- a (/ 1.0 3.0))))))
5.0ms
(/ (/ rand (sqrt 9)) (sqrt (- a (/ 1.0 3.0))))
2.0ms
(sqrt (- a (/ 1.0 3.0)))

series173.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

80.0ms
(/ (/ rand (sqrt 9)) (sqrt (- a (/ 1.0 3.0))))
45.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ (/ rand (sqrt 9)) (sqrt (- a (/ 1.0 3.0))))))
27.0ms
(sqrt (- a (/ 1.0 3.0)))
20.0ms
(- a (/ 1.0 3.0))

simplify5.9s

Counts
100 → 121
Calls

100 calls. Slowest were:

682.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (/ (/ rand (sqrt 9)) (sqrt (- a (/ 1.0 3.0))))))
419.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (/ (/ rand (sqrt 9)) (sqrt (- a (/ 1.0 3.0))))))
396.0ms
(- (* 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))))))

prune1.3s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize18.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ 1 (* 3 (sqrt (- a (/ 1.0 3.0)))))
0.3b
(* 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)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

5.0ms
(* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)
4.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)))
2.0ms
(/ 1 (* 3 (sqrt (- a (/ 1.0 3.0)))))

series177.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

84.0ms
(* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)
40.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)))
28.0ms
(/ 1 (* 3 (sqrt (- a (/ 1.0 3.0)))))
25.0ms
(* 3 (sqrt (- a (/ 1.0 3.0))))

simplify5.1s

Counts
51 → 94
Calls

51 calls. Slowest were:

841.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)))
424.0ms
(+ (* 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))))
339.0ms
(/ 1 (* 3 (sqrt (- (pow a 3) (pow (/ 1.0 3.0) 3)))))

prune938.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes135.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms

end0.0ms

sample6.7s

Algorithm
intervals