Average Error: 0.1 → 0.1
Time: 33.2s
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + {\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\frac{-1}{2}} \cdot rand\right)\]

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 pow1/20.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\color{blue}{{\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\frac{1}{2}}}} \cdot rand\right)\]

  4. Applied pow-flip0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{{\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\left(-\frac{1}{2}\right)}} \cdot rand\right)\]

  5. Simplified0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + {\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\color{blue}{\frac{-1}{2}}} \cdot rand\right)\]

  6. Final simplification0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + {\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\frac{-1}{2}} \cdot rand\right)\]

Reproduce

herbie shell --seed 2019010 +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: 32.5s)Debug log

sample265.0ms

Algorithm
intervals

simplify180.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
180.0ms
(* (- 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

localize50.0ms

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)))

rewrite25.0ms

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×log1p-expm1-u
4×associate-/r/
4×associate-/r*
4×frac-times
4×expm1-log1p-u
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 → 81
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)
5.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))
3.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))

series233.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
97.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
50.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
48.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
37.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify7.4s

Counts
50 → 93
Calls
50 calls:
Slowest
617.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
571.0ms
(* (+ (* a a) (+ (* (/ 1.0 3.0) (/ 1.0 3.0)) (* 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)))))
424.0ms
(* (+ a (/ 1.0 3.0)) (- 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
418.0ms
(+ (* 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))))
416.0ms
(- (* 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))))))

prune882.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

0.4b
(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)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×add-exp-log
7×associate-*l*
7×pow1
6×associate-*r*
6×add-cbrt-cube
4×add-log-exp
4×flip--
4×log1p-expm1-u
4×associate-*r/
4×flip3--
4×frac-times
4×expm1-log1p-u
3×flip-+
3×*-commutative
3×flip3-+
3×distribute-lft-in
3×distribute-rgt-in
2×associate-*l/
2×unpow-prod-down
2×sub-neg
1×cbrt-unprod
1×prod-exp
1×pow-exp
1×pow-prod-down
1×pow-pow
1×pow-to-exp
Counts
4 → 75
Calls
4 calls:
Slowest
6.0ms
(* (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)))
3.0ms
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)
1.0ms
(* 9 (- a (/ 1.0 3.0)))

series230.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
85.0ms
(* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)
60.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
56.0ms
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)
29.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify5.6s

Counts
49 → 87
Calls
49 calls:
Slowest
552.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
465.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
432.0ms
(+ (* 1 1) (- (* (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand) (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)) (* 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand))))
370.0ms
(* (+ a (/ 1.0 3.0)) (- 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
347.0ms
(+ (* 0.041666666666666664 (/ (exp (* -1/2 (- (log 9) (log (/ 1 a))))) (pow a 2))) (+ (exp (* -1/2 (- (log 9) (log (/ 1 a))))) (* 0.16666666666666666 (/ (exp (* -1/2 (- (log 9) (log (/ 1 a))))) a))))

prune1.0s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))
0.2b
(* (pow (- a (/ 1.0 3.0)) -1/2) rand)
0.1b
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
0.0b
(pow (- a (/ 1.0 3.0)) -1/2)

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
12×associate-*l*
10×add-cube-cbrt
10×add-exp-log
10×*-un-lft-identity
10×add-sqr-sqrt
8×add-cbrt-cube
7×pow1
6×unpow-prod-down
4×add-log-exp
4×log1p-expm1-u
4×associate-*r*
4×frac-times
4×expm1-log1p-u
3×flip--
3×flip-+
3×*-commutative
3×prod-exp
3×flip3--
3×flip3-+
2×cbrt-unprod
2×associate-*r/
2×associate-*l/
2×pow-to-exp
1×pow-exp
1×pow-prod-down
1×distribute-lft-in
1×pow-pow
1×distribute-rgt-in
Counts
4 → 77
Calls
4 calls:
Slowest
8.0ms
(* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))
4.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
3.0ms
(* (pow (- a (/ 1.0 3.0)) -1/2) rand)
1.0ms
(pow (- a (/ 1.0 3.0)) -1/2)

series322.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
101.0ms
(* (pow (- a (/ 1.0 3.0)) -1/2) rand)
91.0ms
(* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))
65.0ms
(pow (- a (/ 1.0 3.0)) -1/2)
65.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))

simplify6.7s

Counts
49 → 89
Calls
49 calls:
Slowest
675.0ms
(* (+ a (/ 1.0 3.0)) (- 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
520.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (- (* 1 1) (* (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand)) (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand)))))
517.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
505.0ms
(+ (* 0.16666666666666666 (/ (exp (* -1/2 (- (log -1) (log (/ -1 a))))) a)) (+ (exp (* -1/2 (- (log -1) (log (/ -1 a))))) (* 0.041666666666666664 (/ (exp (* -1/2 (- (log -1) (log (/ -1 a))))) (pow a 2)))))
434.0ms
(- (* (sqrt -3.0) rand) (+ (* 13.5 (/ (* (pow a 2) rand) (sqrt -3.0))) (+ (* 4.5 (/ (* a rand) (sqrt -3.0))) (* 10.125 (/ (* (pow a 2) rand) (pow (sqrt -3.0) 3))))))

prune1.0s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize37.0ms

Local error

Found 3 expressions with local error:

0.4b
(* (pow 9 -1/2) rand)
0.1b
(fma (* (pow 9 -1/2) rand) (pow (- a (/ 1.0 3.0)) 1/2) (- a (/ 1.0 3.0)))
0.0b
(pow (- a (/ 1.0 3.0)) 1/2)

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-exp-log
4×pow1
3×add-log-exp
3×log1p-expm1-u
3×associate-*l*
3×add-cbrt-cube
3×unpow-prod-down
3×expm1-log1p-u
1×unpow1/2
1×fma-udef
1×*-commutative
1×pow-exp
1×pow-pow
1×pow-to-exp
Counts
3 → 39
Calls
3 calls:
Slowest
4.0ms
(* (pow 9 -1/2) rand)
3.0ms
(pow (- a (/ 1.0 3.0)) 1/2)
0.0ms
(fma (* (pow 9 -1/2) rand) (pow (- a (/ 1.0 3.0)) 1/2) (- a (/ 1.0 3.0)))

series193.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
81.0ms
(fma (* (pow 9 -1/2) rand) (pow (- a (/ 1.0 3.0)) 1/2) (- a (/ 1.0 3.0)))
67.0ms
(pow (- a (/ 1.0 3.0)) 1/2)
44.0ms
(* (pow 9 -1/2) rand)

simplify655.0ms

Counts
16 → 48
Calls
16 calls:
Slowest
247.0ms
(- (+ (pow (- 0.3333333333333333) 1/2) (* 1.125 (* (pow a 2) (sqrt -0.3333333333333333)))) (+ (* 1.5 (* a (sqrt -0.3333333333333333))) (* 1/4 (/ (* (pow a 2) (sqrt -0.3333333333333333)) (pow -0.3333333333333333 2)))))
237.0ms
(- (exp (* 1/2 (- (log -1) (log (/ -1 a))))) (+ (* 0.013888888888888888 (/ (exp (* 1/2 (- (log -1) (log (/ -1 a))))) (pow a 2))) (* 0.16666666666666666 (/ (exp (* 1/2 (- (log -1) (log (/ -1 a))))) a))))
84.0ms
(- (pow (/ 1 a) -1/2) (+ (* 0.013888888888888888 (sqrt (/ 1 (pow a 3)))) (* 0.16666666666666666 (sqrt (/ 1 a)))))
50.0ms
(- (+ a (* 1/3 (* (sqrt (- 0.3333333333333333)) rand))) 0.3333333333333333)
7.0ms
(pow (* (cbrt (- a (/ 1.0 3.0))) (cbrt (- a (/ 1.0 3.0)))) 1/2)

prune582.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes131.0ms

Accuracy

0% (0.1b remaining)

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

bsearch3.0ms

end0.0ms

sample6.9s

Algorithm
intervals