Average Error: 0.1 → 0.1
Time: 32.8s
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left({\left(\sqrt{a - \frac{1.0}{3.0}} \cdot \left(\sqrt{a - \frac{1.0}{3.0}} \cdot 9\right)\right)}^{\frac{-1}{2}} \cdot rand + 1\right) \cdot \left(a - \frac{1.0}{3.0}\right)\]

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 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. Using strategy rm

  7. Applied add-sqr-sqrt0.1

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

  8. Applied associate-*r*0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019021 
(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: 32.2s)Debug log

sample363.0ms

Algorithm
intervals
Results
127.0ms919×body80nan
61.0ms289×body80valid

simplify431.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
431.0ms
(* (- 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:

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

rewrite14.0ms

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)))
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))))

series212.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
98.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
47.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
41.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
26.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify6.2s

Counts
50 → 85
Calls
50 calls:
Slowest
447.0ms
(* (+ 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)))))
430.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
366.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
356.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
333.0ms
(- (* 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))))))

prune898.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0b

localize26.0ms

Local error

Found 4 expressions with local error:

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

rewrite18.0ms

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×associate-*r/
4×flip3--
4×frac-times
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 → 67
Calls
4 calls:
Slowest
6.0ms
(* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)
6.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
2.0ms
(* 9 (- a (/ 1.0 3.0)))
2.0ms
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)

series215.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
90.0ms
(* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)
54.0ms
(pow (* 9 (- a (/ 1.0 3.0))) -1/2)
44.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
28.0ms
(* 9 (- a (/ 1.0 3.0)))

simplify5.4s

Counts
49 → 79
Calls
49 calls:
Slowest
606.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (- (* 1 1) (* (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand) (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand))))
445.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
354.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow (* 9 (- a (/ 1.0 3.0))) -1/2) rand)))
347.0ms
(* (+ a (/ 1.0 3.0)) (+ (* 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)))))
310.0ms
(+ (* 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))))

prune800.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0b

localize23.0ms

Local error

Found 4 expressions with local error:

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

rewrite44.0ms

Algorithm
rewrite-expression-head
Rules
14×sqrt-div
12×associate-*r*
10×associate-*r/
9×add-cube-cbrt
9×add-exp-log
9×*-un-lft-identity
9×pow1
9×add-sqr-sqrt
7×flip--
7×flip3--
6×sqrt-prod
6×add-cbrt-cube
5×associate-*l*
4×add-log-exp
4×frac-times
3×*-commutative
2×prod-exp
2×pow-exp
2×associate-*l/
2×pow-prod-down
2×pow-pow
2×unpow-prod-down
1×cbrt-unprod
1×pow-to-exp
Counts
4 → 67
Calls
4 calls:
Slowest
18.0ms
(* (pow (* (* 9 (sqrt (- a (/ 1.0 3.0)))) (sqrt (- a (/ 1.0 3.0)))) -1/2) rand)
11.0ms
(* (* 9 (sqrt (- a (/ 1.0 3.0)))) (sqrt (- a (/ 1.0 3.0))))
9.0ms
(pow (* (* 9 (sqrt (- a (/ 1.0 3.0)))) (sqrt (- a (/ 1.0 3.0)))) -1/2)
3.0ms
(* 9 (sqrt (- a (/ 1.0 3.0))))

series192.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
69.0ms
(* (pow (* (* 9 (sqrt (- a (/ 1.0 3.0)))) (sqrt (- a (/ 1.0 3.0)))) -1/2) rand)
60.0ms
(pow (* (* 9 (sqrt (- a (/ 1.0 3.0)))) (sqrt (- a (/ 1.0 3.0)))) -1/2)
38.0ms
(* 9 (sqrt (- a (/ 1.0 3.0))))
24.0ms
(* (* 9 (sqrt (- a (/ 1.0 3.0)))) (sqrt (- a (/ 1.0 3.0))))

simplify3.6s

Counts
49 → 79
Calls
49 calls:
Slowest
501.0ms
(* (* (* (* 9 (sqrt (- a (/ 1.0 3.0)))) (* 9 (sqrt (- a (/ 1.0 3.0))))) (* 9 (sqrt (- a (/ 1.0 3.0))))) (* (* (sqrt (- a (/ 1.0 3.0))) (sqrt (- a (/ 1.0 3.0)))) (sqrt (- a (/ 1.0 3.0)))))
368.0ms
(* (* 9 (sqrt (- (pow a 3) (pow (/ 1.0 3.0) 3)))) (sqrt (- a (/ 1.0 3.0))))
351.0ms
(* (* 9 (sqrt (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))))) (sqrt (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0)))))
293.0ms
(- (* rand (sqrt -0.3333333333333333)) (+ (* 0.5 (/ (* a rand) (sqrt -0.3333333333333333))) (+ (* 0.125 (/ (* (pow a 2) rand) (pow (sqrt -0.3333333333333333) 3))) (* 1.5 (/ (* (pow a 2) rand) (sqrt -0.3333333333333333))))))
249.0ms
(+ (* 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))))

prune834.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize21.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))
0.3b
(* (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)

rewrite15.0ms

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×associate-*r*
4×frac-times
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 → 69
Calls
4 calls:
Slowest
7.0ms
(* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))
3.0ms
(* (- 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)

series255.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
82.0ms
(* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))
67.0ms
(* (pow (- a (/ 1.0 3.0)) -1/2) rand)
65.0ms
(pow (- a (/ 1.0 3.0)) -1/2)
40.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))

simplify5.2s

Counts
49 → 81
Calls
49 calls:
Slowest
433.0ms
(* (- (* 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)))))
380.0ms
(* (+ a (/ 1.0 3.0)) (- 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
373.0ms
(+ (* 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)))))
358.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (pow 9 -1/2) (* (pow (- a (/ 1.0 3.0)) -1/2) rand))))
348.0ms
(- (* rand (sqrt -0.3333333333333333)) (+ (* 0.5 (/ (* a rand) (sqrt -0.3333333333333333))) (+ (* 0.125 (/ (* (pow a 2) rand) (pow (sqrt -0.3333333333333333) 3))) (* 1.5 (/ (* (pow a 2) rand) (sqrt -0.3333333333333333))))))

prune919.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes120.0ms

Accuracy

0% (0.1b remaining)

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

bsearch4.0ms

end0.0ms

sample6.3s

Algorithm
intervals
Results
4.4s26959×body80nan
1.2s9142×body80valid