Average Error: 0.1 → 0.1
Time: 36.6s
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \frac{rand}{\sqrt{a - \frac{1.0}{3.0}} \cdot 3} + \left(a - \frac{1.0}{3.0}\right)\]

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 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 distribute-rgt-in0.1

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

  8. Applied sqrt-prod0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019018 
(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: 36.0s)Debug log

sample479.0ms

Algorithm
intervals
Results
170.0ms1027×body80nan
56.0ms292×body80valid

simplify398.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
398.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

prune15.0ms

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)
0.2b
(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
Rules
9×associate-*l*
8×add-cube-cbrt
8×associate-*r/
8×*-un-lft-identity
8×add-sqr-sqrt
7×add-exp-log
7×pow1
6×flip--
6×sqrt-div
6×add-cbrt-cube
6×flip3--
4×add-log-exp
4×associate-/r/
4×associate-/r*
4×frac-times
3×flip-+
3×associate-*l/
3×associate-*r*
3×flip3-+
2×pow-flip
2×div-inv
2×sqrt-prod
2×*-commutative
2×pow1/2
1×inv-pow
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×distribute-lft-in
1×frac-2neg
1×clear-num
1×rec-exp
1×distribute-rgt-in
Counts
4 → 73
Calls
4 calls:
Slowest
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)
4.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
1.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

series185.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
85.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
41.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
38.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
21.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify6.2s

Counts
50 → 85
Calls
50 calls:
Slowest
496.0ms
(* (+ 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)))))
382.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
378.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
363.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
342.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (- (* 1 1) (* (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand) (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))

prune617.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×associate-*r/
8×*-un-lft-identity
8×add-sqr-sqrt
6×flip--
6×add-exp-log
6×associate-*r*
6×add-cbrt-cube
6×flip3--
6×pow1
4×add-log-exp
4×sqrt-div
4×associate-/r*
4×frac-times
3×flip-+
3×associate-*l*
3×flip3-+
3×distribute-lft-in
3×distribute-rgt-in
2×sqrt-prod
2×*-commutative
2×associate-/r/
2×associate-*l/
2×sub-neg
1×div-inv
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×pow1/2
1×frac-2neg
1×clear-num
Counts
4 → 71
Calls
4 calls:
Slowest
5.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
2.0ms
(* 9 (- a (/ 1.0 3.0)))
1.0ms
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
1.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

series171.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
70.0ms
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
48.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
35.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))
17.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify4.6s

Counts
52 → 83
Calls
52 calls:
Slowest
440.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
407.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ (pow 1 3) (pow (/ rand (sqrt (* 9 (- a (/ 1.0 3.0))))) 3)))
317.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
275.0ms
(- (* 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))))))
264.0ms
(* (+ 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)))))))))

prune778.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

10.1b
(* (/ rand (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
0.2b
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
0.2b
(sqrt (* 9 (- a (/ 1.0 3.0))))
0.0b
(* 9 (- a (/ 1.0 3.0)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
10×associate-*r/
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×flip--
6×add-exp-log
6×associate-*l*
6×associate-*r*
6×sqrt-div
6×add-cbrt-cube
6×flip3--
6×pow1
4×add-log-exp
4×associate-/r/
4×distribute-lft-in
4×sub-neg
4×associate-/r*
4×distribute-rgt-in
2×div-inv
2×sqrt-prod
2×*-commutative
2×frac-times
1×cbrt-unprod
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×pow1/2
1×frac-2neg
1×clear-num
Counts
4 → 73
Calls
4 calls:
Slowest
5.0ms
(* (/ rand (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
1.0ms
(* 9 (- a (/ 1.0 3.0)))
1.0ms
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
1.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

series190.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
73.0ms
(* (/ rand (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
68.0ms
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
27.0ms
(* 9 (- a (/ 1.0 3.0)))
21.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify4.5s

Counts
51 → 85
Calls
51 calls:
Slowest
733.0ms
(/ rand (sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3)))))
393.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
368.0ms
(* rand (- (pow a 3) (pow (/ 1.0 3.0) 3)))
366.0ms
(- (* 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))))))
307.0ms
(- (+ (* 1/3 (* (sqrt (- 0.3333333333333333)) rand)) (* 1/6 (/ (* a rand) (sqrt (- 0.3333333333333333))))) (* 1/24 (/ (* (pow a 2) rand) (pow (sqrt (- 0.3333333333333333)) 3))))

prune836.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 4 expressions with local error:

10.1b
(* (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
0.3b
(* 3 (sqrt (- a (/ 1.0 3.0))))
0.2b
(/ rand (* 3 (sqrt (- a (/ 1.0 3.0)))))
0.0b
(+ (* 1 (- a (/ 1.0 3.0))) (* (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0))))

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
22×associate-*r/
14×flip--
14×flip3--
10×frac-add
9×associate-*r*
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-log-exp
6×add-exp-log
6×associate-*l*
6×sqrt-div
6×add-cbrt-cube
6×distribute-lft-in
6×sub-neg
6×pow1
6×frac-times
6×distribute-rgt-in
4×associate-/r/
4×associate-+r+
4×associate-+l+
3×sqrt-prod
3×associate-*l/
2×div-inv
2×*-commutative
1×flip-+
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×distribute-rgt-out
1×flip3-+
1×frac-2neg
1×sum-log
1×associate-/r*
1×clear-num
1×+-commutative
Counts
4 → 88
Calls
4 calls:
Slowest
5.0ms
(* (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
5.0ms
(+ (* 1 (- a (/ 1.0 3.0))) (* (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0))))
2.0ms
(* 3 (sqrt (- a (/ 1.0 3.0))))
1.0ms
(/ rand (* 3 (sqrt (- a (/ 1.0 3.0)))))

series217.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
68.0ms
(+ (* 1 (- a (/ 1.0 3.0))) (* (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0))))
61.0ms
(/ rand (* 3 (sqrt (- a (/ 1.0 3.0)))))
61.0ms
(* (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
27.0ms
(* 3 (sqrt (- a (/ 1.0 3.0))))

simplify8.8s

Counts
75 → 100
Calls
75 calls:
Slowest
563.0ms
(* (+ (* a a) (+ (* (/ 1.0 3.0) (/ 1.0 3.0)) (* a (/ 1.0 3.0)))) (+ (* a a) (+ (* (/ 1.0 3.0) (/ 1.0 3.0)) (* a (/ 1.0 3.0)))))
458.0ms
(+ (* (* 1 (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0)))) (* 3 (sqrt (- a (/ 1.0 3.0))))) (* (+ a (/ 1.0 3.0)) (* rand (- a (/ 1.0 3.0)))))
383.0ms
(* rand (- (pow a 3) (pow (/ 1.0 3.0) 3)))
349.0ms
(- (* 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))))))
343.0ms
(* (/ rand (* 3 (sqrt (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))

prune1.0s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

regimes176.0ms

Accuracy

0% (0.1b remaining)

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

bsearch5.0ms

end0.0ms

sample6.7s

Algorithm
intervals
Results
4.0s27255×body80nan
1.6s9040×body80valid