Average Error: 0.1 → 0.1
Time: 33.1s
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} + \left(a - \frac{1.0}{3.0}\right)\]

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019016 
(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: 32.6s)Debug log

sample308.0ms

Algorithm
intervals
Results
1215×(pre true 80)
911×(body nan 80)
304×(body real 80)

simplify454.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
454.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.2b

localize22.0ms

Local error

Found 4 expressions with local error:

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

rewrite10.0ms

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
4.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
3.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
2.0ms
(/ 1 (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
71.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
44.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
35.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
22.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify5.9s

Counts
50 → 85
Calls
50 calls:
Slowest
489.0ms
(* (+ 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)))))
461.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
362.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
328.0ms
(/ 1 (sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3)))))
316.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

prune842.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite9.0ms

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)))))))
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))))

series125.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
51.0ms
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
36.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
21.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))
17.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify4.4s

Counts
52 → 83
Calls
52 calls:
Slowest
375.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))
337.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
323.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ (pow 1 3) (pow (/ rand (sqrt (* 9 (- a (/ 1.0 3.0))))) 3)))
289.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
270.0ms
(- (* 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))))))

prune570.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

6.0b
(* (/ rand (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
0.3b
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
0.1b
(sqrt (* 9 (- a (/ 1.0 3.0))))
0.0b
(* 9 (- a (/ 1.0 3.0)))

rewrite11.0ms

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
7.0ms
(* (/ 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))))

series142.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
51.0ms
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
43.0ms
(* (/ rand (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
27.0ms
(* 9 (- a (/ 1.0 3.0)))
21.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify3.2s

Counts
51 → 85
Calls
51 calls:
Slowest
366.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
304.0ms
(/ rand (sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3)))))
265.0ms
(* rand (- (pow a 3) (pow (/ 1.0 3.0) 3)))
236.0ms
(- (* 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))))))
204.0ms
(- (+ (* 1/3 (* (sqrt (- 0.3333333333333333)) rand)) (* 1/6 (/ (* a rand) (sqrt (- 0.3333333333333333))))) (* 1/24 (/ (* (pow a 2) rand) (pow (sqrt (- 0.3333333333333333)) 3))))

prune578.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 4 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
15×associate-*r/
10×distribute-lft-in
10×distribute-rgt-in
9×add-exp-log
7×flip--
7×add-cube-cbrt
7×flip3--
7×*-un-lft-identity
7×pow1
7×add-sqr-sqrt
6×associate-*l*
6×sqrt-div
6×add-cbrt-cube
6×sub-neg
4×add-log-exp
4×associate-/r/
4×associate-*r*
4×associate-/r*
4×frac-times
2×pow-flip
2×div-inv
2×sqrt-prod
2×*-commutative
2×prod-exp
2×associate-*l/
2×pow1/2
2×rec-exp
1×inv-pow
1×cbrt-unprod
1×pow-prod-down
1×frac-2neg
1×clear-num
Counts
4 → 83
Calls
4 calls:
Slowest
5.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
4.0ms
(* rand (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0))))
2.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
1.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

series162.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
52.0ms
(* rand (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0))))
45.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
41.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
24.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify7.0s

Counts
59 → 95
Calls
59 calls:
Slowest
550.0ms
(* (* (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))) (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))) (* (* (- a (/ 1.0 3.0)) (- a (/ 1.0 3.0))) (- a (/ 1.0 3.0))))
470.0ms
(* 1 (- (pow a 3) (pow (/ 1.0 3.0) 3)))
409.0ms
(* (- (/ 1.0 3.0)) (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))))
404.0ms
(* (- (/ 1.0 3.0)) (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))))
379.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) (- (/ 1.0 3.0)))

prune975.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes143.0ms

Accuracy

0% (0.1b remaining)

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

bsearch5.0ms

end0.0ms

sample7.4s

Algorithm
intervals
Results
36442×(pre true 80)
27308×(body nan 80)
9134×(body real 80)