Average Error: 0.1 → 0.1
Time: 40.7s
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 +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: 40.1s)Debug log

sample336.0ms

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

simplify580.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
580.0ms
(* (- 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

localize29.0ms

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)))

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)))
8.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
4.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))
3.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))

series194.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
86.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)
47.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
37.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
24.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify7.6s

Counts
50 → 93
Calls
50 calls:
Slowest
659.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
504.0ms
(+ (* 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))))
472.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))
439.0ms
(* (- (* 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))))
412.0ms
(* (+ a (/ 1.0 3.0)) (- 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand)))

prune1.0s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

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)))

rewrite21.0ms

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×log1p-expm1-u
4×sqrt-div
4×associate-/r*
4×frac-times
4×expm1-log1p-u
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 → 79
Calls
4 calls:
Slowest
11.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
4.0ms
(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)))))

series176.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
70.0ms
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
52.0ms
(* (- a (/ 1.0 3.0)) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
31.0ms
(* 9 (- a (/ 1.0 3.0)))
22.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify5.8s

Counts
52 → 91
Calls
52 calls:
Slowest
634.0ms
(* (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0))) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
484.0ms
(* (+ 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)))))))))
400.0ms
(* (- (pow a 3) (pow (/ 1.0 3.0) 3)) (+ 1 (/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))))
396.0ms
(- (* 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))))))
370.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))

prune896.0ms

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)))

rewrite10.0ms

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×log1p-expm1-u
4×associate-/r/
4×distribute-lft-in
4×sub-neg
4×associate-/r*
4×expm1-log1p-u
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 → 81
Calls
4 calls:
Slowest
5.0ms
(* (/ rand (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
2.0ms
(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)))))

series170.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
58.0ms
(* (/ rand (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
52.0ms
(/ rand (sqrt (* 9 (- a (/ 1.0 3.0)))))
30.0ms
(* 9 (- a (/ 1.0 3.0)))
29.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify5.0s

Counts
51 → 93
Calls
51 calls:
Slowest
422.0ms
(* rand (- (pow a 3) (pow (/ 1.0 3.0) 3)))
398.0ms
(- (* 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))))))
329.0ms
(/ rand (sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3)))))
327.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
327.0ms
(sqrt (* 9 (- (* a a) (* (/ 1.0 3.0) (/ 1.0 3.0)))))

prune1.0s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize11.0ms

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))))

rewrite29.0ms

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×log1p-expm1-u
4×associate-/r/
4×associate-*r*
4×associate-/r*
4×frac-times
4×expm1-log1p-u
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 → 91
Calls
4 calls:
Slowest
11.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
9.0ms
(* rand (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0))))
4.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))
3.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))

series178.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
60.0ms
(* rand (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0))))
51.0ms
(/ 1 (sqrt (* 9 (- a (/ 1.0 3.0)))))
44.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
23.0ms
(sqrt (* 9 (- a (/ 1.0 3.0))))

simplify8.3s

Counts
59 → 103
Calls
59 calls:
Slowest
782.0ms
(* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) (- a (/ 1.0 3.0)))
634.0ms
(* (* (* (/ 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))))
436.0ms
(* 1 (- (pow a 3) (pow (/ 1.0 3.0) 3)))
420.0ms
(sqrt (* 9 (- (pow a 3) (pow (/ 1.0 3.0) 3))))
395.0ms
(- (+ (* 1/3 (* (sqrt (- 0.3333333333333333)) rand)) (* 1/6 (/ (* a rand) (sqrt (- 0.3333333333333333))))) (* 1/24 (/ (* (pow a 2) rand) (pow (sqrt (- 0.3333333333333333)) 3))))

prune1.1s

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes139.0ms

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)