Average Error: 0.0 → 0.0
Time: 33.1s
Precision: 64
Internal Precision: 128
\[\frac{-\left(f + n\right)}{f - n}\]
↓
\[\log \left(\sqrt{e^{-\frac{n + f}{f - n}}}\right) + \log \left(\sqrt{e^{-\frac{n + f}{f - n}}}\right)\]

Error

Bits error versus f

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\frac{-\left(f + n\right)}{f - n}\]
  2. Using strategy rm

  3. Applied add-log-exp0.0

    \[\leadsto \color{blue}{\log \left(e^{\frac{-\left(f + n\right)}{f - n}}\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.0

    \[\leadsto \log \color{blue}{\left(\sqrt{e^{\frac{-\left(f + n\right)}{f - n}}} \cdot \sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}\right)}\]

  6. Applied log-prod0.0

    \[\leadsto \color{blue}{\log \left(\sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}\right) + \log \left(\sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}\right)}\]

  7. Final simplification0.0

    \[\leadsto \log \left(\sqrt{e^{-\frac{n + f}{f - n}}}\right) + \log \left(\sqrt{e^{-\frac{n + f}{f - n}}}\right)\]

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (f n)
  :name "subtraction fraction"
  (/ (- (+ f n)) (- f n)))

Details

Time bar (total: 32.8s)Debug log

sample117.0ms

Algorithm
intervals

simplify42.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
41.0ms
(/ (- (+ f n)) (- f n))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize56.0ms

Local error

Found 1 expressions with local error:

0.0b
(/ (- (+ f n)) (- f n))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
30×times-frac
24×add-cube-cbrt
24×*-un-lft-identity
24×add-sqr-sqrt
12×distribute-rgt-neg-in
12×distribute-lft-neg-in
10×associate-/l*
4×neg-mul-1
3×add-exp-log
3×add-cbrt-cube
3×associate-/r*
2×associate-/l/
2×distribute-neg-frac
2×associate-/r/
1×add-log-exp
1×flip--
1×div-inv
1×log1p-expm1-u
1×flip-+
1×distribute-frac-neg
1×div-exp
1×div-sub
1×flip3--
1×flip3-+
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×pow1
1×expm1-log1p-u
1×neg-sub0
Counts
1 → 63
Calls
1 calls:
Slowest
12.0ms
(/ (- (+ f n)) (- f n))

series102.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
102.0ms
(/ (- (+ f n)) (- f n))

simplify1.7s

Counts
81 → 66
Calls
81 calls:
Slowest
249.0ms
(/ (* (* (- (+ f n)) (- (+ f n))) (- (+ f n))) (* (* (- f n) (- f n)) (- f n)))
229.0ms
(- (- (* f f) (* n n)))
70.0ms
(/ (* (cbrt (- (+ f n))) (cbrt (- (+ f n)))) 1)
63.0ms
(/ (sqrt (- (+ f n))) (* (cbrt (- f n)) (cbrt (- f n))))
52.0ms
(/ (- (* (cbrt (+ f n)) (cbrt (+ f n)))) (* (cbrt (- f n)) (cbrt (- f n))))

prune536.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

localize7.0ms

Local error

Found 3 expressions with local error:

0.0b
(exp (/ (- (+ f n)) (- f n)))
0.0b
(/ (- (+ f n)) (- f n))
0.0b
(log (exp (/ (- (+ f n)) (- f n))))

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
90×times-frac
72×exp-prod
67×add-cube-cbrt
67×*-un-lft-identity
67×add-sqr-sqrt
37×log-pow
30×distribute-rgt-neg-in
30×distribute-lft-neg-in
10×associate-/l*
10×neg-mul-1
6×associate-/r/
5×add-exp-log
5×add-cbrt-cube
4×add-log-exp
4×distribute-frac-neg
4×pow1
3×flip--
3×div-inv
3×log1p-expm1-u
3×div-sub
3×flip3--
3×associate-/r*
3×log-prod
3×expm1-log1p-u
3×exp-neg
3×neg-sub0
2×associate-/l/
2×distribute-neg-frac
2×exp-diff
2×log-div
1×rem-exp-log
1×flip-+
1×log-rec
1×rem-log-exp
1×div-exp
1×flip3-+
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
3 → 164
Calls
3 calls:
Slowest
12.0ms
(exp (/ (- (+ f n)) (- f n)))
12.0ms
(log (exp (/ (- (+ f n)) (- f n))))
9.0ms
(/ (- (+ f n)) (- f n))

series547.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
412.0ms
(/ (- (+ f n)) (- f n))
97.0ms
(exp (/ (- (+ f n)) (- f n)))
38.0ms
(log (exp (/ (- (+ f n)) (- f n))))

simplify2.8s

Counts
168 → 173
Calls
168 calls:
Slowest
259.0ms
(- (- (* f f) (* n n)))
152.0ms
(exp (/ (- (+ f n)) (- (* f f) (* n n))))
111.0ms
(/ (* (* (- (+ f n)) (- (+ f n))) (- (+ f n))) (* (* (- f n) (- f n)) (- f n)))
77.0ms
(exp (/ (- (* (cbrt (+ f n)) (cbrt (+ f n)))) (* (cbrt (- f n)) (cbrt (- f n)))))
63.0ms
(exp (/ (- (+ f n)) (- (pow f 3) (pow n 3))))

prune1.6s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize18.0ms

Local error

Found 4 expressions with local error:

0.0b
(exp (/ (- (+ f n)) (- f n)))
0.0b
(exp (/ (- (+ f n)) (- f n)))
0.0b
(/ (- (+ f n)) (- f n))
0.0b
(/ (- (+ f n)) (- f n))

rewrite47.0ms

Algorithm
rewrite-expression-head
Rules
120×times-frac
90×add-cube-cbrt
90×*-un-lft-identity
90×add-sqr-sqrt
72×exp-prod
42×distribute-rgt-neg-in
42×distribute-lft-neg-in
20×associate-/l*
14×neg-mul-1
8×associate-/r/
8×add-exp-log
8×add-cbrt-cube
6×add-log-exp
6×associate-/r*
4×associate-/l/
4×distribute-neg-frac
4×flip--
4×div-inv
4×log1p-expm1-u
4×distribute-frac-neg
4×div-sub
4×flip3--
4×pow1
4×expm1-log1p-u
4×neg-sub0
2×rem-exp-log
2×flip-+
2×div-exp
2×exp-diff
2×flip3-+
2×frac-2neg
2×clear-num
2×cbrt-undiv
2×exp-neg
Counts
4 → 222
Calls
4 calls:
Slowest
12.0ms
(/ (- (+ f n)) (- f n))
9.0ms
(exp (/ (- (+ f n)) (- f n)))
9.0ms
(/ (- (+ f n)) (- f n))
9.0ms
(exp (/ (- (+ f n)) (- f n)))

series228.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
92.0ms
(exp (/ (- (+ f n)) (- f n)))
73.0ms
(exp (/ (- (+ f n)) (- f n)))
32.0ms
(/ (- (+ f n)) (- f n))
31.0ms
(/ (- (+ f n)) (- f n))

simplify4.4s

Counts
244 → 234
Calls
244 calls:
Slowest
267.0ms
(- (- (* f f) (* n n)))
252.0ms
(- (- (* f f) (* n n)))
145.0ms
(exp (/ (- (+ f n)) (- (* f f) (* n n))))
134.0ms
(exp (/ (- (+ f n)) (- (* f f) (* n n))))
104.0ms
(/ (* (* (- (+ f n)) (- (+ f n))) (- (+ f n))) (* (* (- f n) (- f n)) (- f n)))

prune3.5s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize37.0ms

Local error

Found 4 expressions with local error:

29.6b
(/ (- (+ f n)) (- (* f f) (* n n)))
2.2b
(- (* f f) (* n n))
0.1b
(* (/ (- (+ f n)) (- (* f f) (* n n))) (+ f n))
0.0b
(exp (* (/ (- (+ f n)) (- (* f f) (* n n))) (+ f n)))

rewrite28.0ms

Algorithm
rewrite-expression-head
Rules
80×times-frac
54×add-cube-cbrt
54×*-un-lft-identity
54×add-sqr-sqrt
46×associate-*l*
27×distribute-rgt-neg-in
27×distribute-lft-neg-in
22×difference-of-squares
11×add-exp-log
11×add-cbrt-cube
10×associate-/l*
9×neg-mul-1
8×add-log-exp
6×pow1
4×log1p-expm1-u
4×associate-/r/
4×associate-/r*
4×expm1-log1p-u
3×flip--
3×flip-+
3×distribute-frac-neg
3×associate-*r*
3×flip3--
3×flip3-+
2×associate-/l/
2×distribute-neg-frac
2×div-inv
2×cbrt-unprod
2×associate-*r/
2×prod-exp
2×exp-sum
2×div-exp
2×distribute-lft-in
2×cbrt-undiv
2×frac-times
2×distribute-rgt-in
2×distribute-lft-neg-out
1×rem-exp-log
1×*-commutative
1×associate-*l/
1×pow-prod-down
1×prod-diff
1×div-sub
1×diff-log
1×fma-neg
1×exp-to-pow
1×frac-2neg
1×sub-neg
1×clear-num
1×exp-prod
1×exp-neg
1×neg-sub0
Counts
4 → 178
Calls
4 calls:
Slowest
9.0ms
(* (/ (- (+ f n)) (- (* f f) (* n n))) (+ f n))
7.0ms
(/ (- (+ f n)) (- (* f f) (* n n)))
6.0ms
(- (* f f) (* n n))
3.0ms
(exp (* (/ (- (+ f n)) (- (* f f) (* n n))) (+ f n)))

series233.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
100.0ms
(exp (* (/ (- (+ f n)) (- (* f f) (* n n))) (+ f n)))
68.0ms
(* (/ (- (+ f n)) (- (* f f) (* n n))) (+ f n))
37.0ms
(/ (- (+ f n)) (- (* f f) (* n n)))
28.0ms
(- (* f f) (* n n))

simplify14.5s

Counts
184 → 190
Calls
184 calls:
Slowest
533.0ms
(* (- (* f f) (* n n)) (+ (* f f) (- (* n n) (* f n))))
456.0ms
(* (/ (+ f n) (- (* f f) (* n n))) (+ f n))
347.0ms
(* (/ (* (* (- (+ f n)) (- (+ f n))) (- (+ f n))) (* (* (- (* f f) (* n n)) (- (* f f) (* n n))) (- (* f f) (* n n)))) (* (* (+ f n) (+ f n)) (+ f n)))
341.0ms
(* (* (* (/ (- (+ f n)) (- (* f f) (* n n))) (/ (- (+ f n)) (- (* f f) (* n n)))) (/ (- (+ f n)) (- (* f f) (* n n)))) (* (* (+ f n) (+ f n)) (+ f n)))
279.0ms
(* (- (+ f n)) (+ f n))

prune1.4s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes81.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample772.0ms

Algorithm
intervals