Average Error: 0.1 → 0.1
Time: 36.3s
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(\frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} + 1\right) \cdot \left(a - \frac{1.0}{3.0}\right)\]

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 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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019010 
(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: 35.8s)Debug log

sample364.0ms

Algorithm
intervals

simplify415.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
415.0ms
(* (- 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: 0.1b

localize31.0ms

Local error

Found 4 expressions with local error:

0.4b
(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)))

rewrite20.0ms

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

series234.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
86.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
58.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
53.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
37.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify6.5s

Counts
50 → 85
Calls
50 calls:
Slowest
510.0ms
(* (+ 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)))))
502.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
442.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
384.0ms
(- (* 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))))))
331.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))

prune901.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 4 expressions with local error:

0.4b
(sqrt (* 9 (- a (/ 1.0 3.0))))
0.2b
(/ 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)))

rewrite17.0ms

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
10.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
3.0ms
(* 9 (- a (/ 1.0 3.0)))
2.0ms
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
1.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

series182.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
61.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
56.0ms
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
35.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))
29.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify4.9s

Counts
52 → 83
Calls
52 calls:
Slowest
456.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ (pow 1 3) (pow (/ rand (sqrt (* 9 (- a (/ 1.0 3.0))))) 3)))
375.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
333.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
306.0ms
(* (+ 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)))))))))
290.0ms
(- (* 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))))))

prune847.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize23.0ms

Local error

Found 4 expressions with local error:

1.5b
(/ 1 (/ (sqrt (* 9 (- a (/ 1.0 3.0)))) rand))
0.4b
(sqrt (* 9 (- a (/ 1.0 3.0))))
0.2b
(/ (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))))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×add-exp-log
7×pow1
6×associate-*r/
6×add-cbrt-cube
5×flip--
5×flip3--
4×add-log-exp
4×associate-/l*
4×sqrt-div
4×associate-/r*
4×frac-times
3×div-inv
3×flip-+
3×associate-*l*
3×associate-*r*
3×flip3-+
2×associate-/l/
2×sqrt-prod
2×associate-*l/
2×frac-2neg
2×clear-num
1×inv-pow
1×pow-flip
1×cbrt-unprod
1×*-commutative
1×associate-/r/
1×prod-exp
1×pow-prod-down
1×pow1/2
1×distribute-lft-in
1×rec-exp
1×distribute-rgt-in
Counts
4 → 72
Calls
4 calls:
Slowest
8.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ 1 (/ (sqrt (* 9 (- a (/ 1.0 3.0)))) rand))))
5.0ms
(/ (sqrt (* 9 (- a (/ 1.0 3.0)))) rand)
2.0ms
(/ 1 (/ (sqrt (* 9 (- a (/ 1.0 3.0)))) rand))
1.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

series254.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
91.0ms
(/ 1 (/ (sqrt (* 9 (- a (/ 1.0 3.0)))) rand))
86.0ms
(/ (sqrt (* 9 (- a (/ 1.0 3.0)))) rand)
57.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ 1 (/ (sqrt (* 9 (- a (/ 1.0 3.0)))) rand))))
21.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify6.3s

Counts
47 → 84
Calls
47 calls:
Slowest
477.0ms
(- (+ (* 3 (sqrt (- 0.3333333333333333))) (* 3/2 (/ a (sqrt (- 0.3333333333333333))))) (* 3/8 (/ (pow a 2) (pow (sqrt (- 0.3333333333333333)) 3))))
445.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (/ 1 (/ (sqrt (* 9 (- a (/ 1.0 3.0)))) rand))))
413.0ms
(* (+ 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))))))
388.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
365.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))

prune842.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize16.0ms

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)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
9×add-cube-cbrt
9×associate-*l*
9×associate-*r*
9×*-un-lft-identity
9×add-sqr-sqrt
8×associate-*r/
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×frac-times
3×flip-+
3×sqrt-prod
3×*-commutative
3×associate-*l/
3×flip3-+
2×div-inv
1×inv-pow
1×pow-flip
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×distribute-lft-in
1×frac-2neg
1×associate-/r*
1×clear-num
1×rec-exp
1×distribute-rgt-in
Counts
4 → 74
Calls
4 calls:
Slowest
7.0ms
(* (/ 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)))))
2.0ms
(* 3 (sqrt (- a (/ 1.0 3.0))))

series201.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
86.0ms
(* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)
46.0ms
(/ 1 (* 3 (sqrt (- a (/ 1.0 3.0)))))
40.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)))
28.0ms
(* 3 (sqrt (- a (/ 1.0 3.0))))

simplify5.6s

Counts
51 → 86
Calls
51 calls:
Slowest
731.0ms
(* (- (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))))
628.0ms
(* (+ a (/ 1.0 3.0)) (- 1 (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)))
627.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (/ 1 (* 3 (sqrt (- a (/ 1.0 3.0))))) rand)))
326.0ms
(- (* 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))))))
297.0ms
(/ 1 (* 3 (sqrt (- (pow a 3) (pow (/ 1.0 3.0) 3)))))

prune781.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

regimes125.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms

end0.0ms

sample7.2s

Algorithm
intervals