Average Error: 0.1 → 0.1
Time: 31.9s
Precision: 64
Internal Precision: 128
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(\frac{m - {m}^{3}}{\left(1 + m\right) \cdot v} - 1\right) \cdot \left(1 - m\right)\]

Error

Bits error versus m

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
  2. Using strategy rm

  3. Applied flip--0.1

    \[\leadsto \left(\frac{m \cdot \color{blue}{\frac{1 \cdot 1 - m \cdot m}{1 + m}}}{v} - 1\right) \cdot \left(1 - m\right)\]

  4. Applied associate-*r/0.1

    \[\leadsto \left(\frac{\color{blue}{\frac{m \cdot \left(1 \cdot 1 - m \cdot m\right)}{1 + m}}}{v} - 1\right) \cdot \left(1 - m\right)\]

  5. Applied associate-/l/0.1

    \[\leadsto \left(\color{blue}{\frac{m \cdot \left(1 \cdot 1 - m \cdot m\right)}{v \cdot \left(1 + m\right)}} - 1\right) \cdot \left(1 - m\right)\]

  6. Simplified0.1

    \[\leadsto \left(\frac{\color{blue}{(\left(m \cdot m\right) \cdot \left(-m\right) + m)_*}}{v \cdot \left(1 + m\right)} - 1\right) \cdot \left(1 - m\right)\]

  7. Taylor expanded around inf 0.1

    \[\leadsto \left(\frac{\color{blue}{m - {m}^{3}}}{v \cdot \left(1 + m\right)} - 1\right) \cdot \left(1 - m\right)\]

  8. Final simplification0.1

    \[\leadsto \left(\frac{m - {m}^{3}}{\left(1 + m\right) \cdot v} - 1\right) \cdot \left(1 - m\right)\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :pre (and (< 0 m) (< 0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))

Details

Time bar (total: 31.2s)Debug log

sample211.0ms

Algorithm
intervals

simplify515.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

514.0ms
(* (- (/ (* m (- 1 m)) v) 1) (- 1 m))

prune22.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize29.0ms

Local error

Found 2 expressions with local error:

0.1b
(* (fma (- 1 m) (/ m v) -1) (- 1 m))
0.1b
(fma (- 1 m) (/ m v) -1)

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
2 → 35
Calls

2 calls. Slowest were:

11.0ms
(* (fma (- 1 m) (/ m v) -1) (- 1 m))
0.0ms
(fma (- 1 m) (/ m v) -1)

series77.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

57.0ms
(* (fma (- 1 m) (/ m v) -1) (- 1 m))
20.0ms
(fma (- 1 m) (/ m v) -1)

simplify5.1s

Counts
20 → 41
Calls

20 calls. Slowest were:

1.0s
(* (- m) (fma (- 1 m) (/ m v) -1))
948.0ms
(* (- m) (fma (- 1 m) (/ m v) -1))
724.0ms
(* (* (* (fma (- 1 m) (/ m v) -1) (fma (- 1 m) (/ m v) -1)) (fma (- 1 m) (/ m v) -1)) (* (* (- 1 m) (- 1 m)) (- 1 m)))

prune294.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 3 expressions with local error:

0.2b
(fma (/ m v) (* m (+ -1 m)) m)
0.1b
(fma (- 1 m) (/ m v) -1)
0.0b
(+ (* 1 (fma (- 1 m) (/ m v) -1)) (fma (/ m v) (* m (+ -1 m)) m))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 38
Calls

3 calls. Slowest were:

8.0ms
(+ (* 1 (fma (- 1 m) (/ m v) -1)) (fma (/ m v) (* m (+ -1 m)) m))
0.0ms
(fma (/ m v) (* m (+ -1 m)) m)
0.0ms
(fma (- 1 m) (/ m v) -1)

series64.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

27.0ms
(fma (/ m v) (* m (+ -1 m)) m)
23.0ms
(+ (* 1 (fma (- 1 m) (/ m v) -1)) (fma (/ m v) (* m (+ -1 m)) m))
14.0ms
(fma (- 1 m) (/ m v) -1)

simplify4.6s

Counts
13 → 47
Calls

13 calls. Slowest were:

2.5s
(+ (fma (- 1 m) (/ m v) -1) (fma (/ m v) (* m (+ -1 m)) m))
326.0ms
(* 1 (* (- 1 m) (/ m v)))
315.0ms
(- (+ m (/ (pow m 3) v)) (* 2 (/ (pow m 2) v)))

prune552.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 3 expressions with local error:

0.1b
(* (- (/ (* m (- 1 m)) v) 1) (- 1 m))
0.1b
(/ (* m (- 1 m)) v)
0.0b
(- (/ (* m (- 1 m)) v) 1)

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
3 → 66
Calls

3 calls. Slowest were:

12.0ms
(* (- (/ (* m (- 1 m)) v) 1) (- 1 m))
4.0ms
(- (/ (* m (- 1 m)) v) 1)
4.0ms
(/ (* m (- 1 m)) v)

series67.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

28.0ms
(/ (* m (- 1 m)) v)
21.0ms
(* (- (/ (* m (- 1 m)) v) 1) (- 1 m))
18.0ms
(- (/ (* m (- 1 m)) v) 1)

simplify8.3s

Counts
41 → 75
Calls

41 calls. Slowest were:

750.0ms
(* (- m) (- (/ (* m (- 1 m)) v) 1))
678.0ms
(* (- m) (- (/ (* m (- 1 m)) v) 1))
645.0ms
(* (+ (/ (* m (- 1 m)) v) 1) (+ 1 m))

prune712.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize51.0ms

Local error

Found 4 expressions with local error:

0.1b
(/ (fma (* m m) (- m) m) (* v (+ 1 m)))
0.1b
(* (- (/ (fma (* m m) (- m) m) (* v (+ 1 m))) 1) (- 1 m))
0.0b
(fma (* m m) (- m) m)
0.0b
(- (/ (fma (* m m) (- m) m) (* v (+ 1 m))) 1)

rewrite53.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

33.0ms
(* (- (/ (fma (* m m) (- m) m) (* v (+ 1 m))) 1) (- 1 m))
10.0ms
(- (/ (fma (* m m) (- m) m) (* v (+ 1 m))) 1)
8.0ms
(/ (fma (* m m) (- m) m) (* v (+ 1 m)))

series177.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

63.0ms
(fma (* m m) (- m) m)
43.0ms
(- (/ (fma (* m m) (- m) m) (* v (+ 1 m))) 1)
37.0ms
(/ (fma (* m m) (- m) m) (* v (+ 1 m)))
33.0ms
(* (- (/ (fma (* m m) (- m) m) (* v (+ 1 m))) 1) (- 1 m))

simplify7.2s

Counts
61 → 102
Calls

61 calls. Slowest were:

989.0ms
(* (* (* (- (/ (fma (* m m) (- m) m) (* v (+ 1 m))) 1) (- (/ (fma (* m m) (- m) m) (* v (+ 1 m))) 1)) (- (/ (fma (* m m) (- m) m) (* v (+ 1 m))) 1)) (* (* (- 1 m) (- 1 m)) (- 1 m)))
649.0ms
(* (+ (/ (fma (* m m) (- m) m) (* v (+ 1 m))) 1) (+ 1 m))
417.0ms
(* (- (pow (/ (fma (* m m) (- m) m) (* v (+ 1 m))) 3) (pow 1 3)) (- (pow 1 3) (pow m 3)))

prune1.0s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.0b

regimes89.0ms

Accuracy

0% (0.1b remaining)

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

bsearch2.0ms

end0.0ms

sample2.0s

Algorithm
intervals