Average Error: 2.1 → 2.0
Time: 28.0s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{\sqrt{{k}^{m}} \cdot \left(\sqrt{{k}^{m}} \cdot a\right)}{1 + \left(k + 10\right) \cdot k}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.1

    \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

  2. Initial simplification2.0

    \[\leadsto \frac{{k}^{m} \cdot a}{1 + k \cdot \left(k + 10\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt2.1

    \[\leadsto \frac{\color{blue}{\left(\sqrt{{k}^{m}} \cdot \sqrt{{k}^{m}}\right)} \cdot a}{1 + k \cdot \left(k + 10\right)}\]

  5. Applied associate-*l*2.0

    \[\leadsto \frac{\color{blue}{\sqrt{{k}^{m}} \cdot \left(\sqrt{{k}^{m}} \cdot a\right)}}{1 + k \cdot \left(k + 10\right)}\]

  6. Final simplification2.0

    \[\leadsto \frac{\sqrt{{k}^{m}} \cdot \left(\sqrt{{k}^{m}} \cdot a\right)}{1 + \left(k + 10\right) \cdot k}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))

Details

Time bar (total: 22.7s)Debug log

start166.0ms

Algorithm
intervals

setup68.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 2.3b

localize18.0ms

Local error

Found 3 expressions with local error:

2.3b
(/ (* (pow k m) a) (+ 1 (* k (+ k 10))))
0.0b
(* k (+ k 10))
0.0b
(* (pow k m) a)

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 47
Calls

3 calls. Slowest were:

5.0ms
(/ (* (pow k m) a) (+ 1 (* k (+ k 10))))
2.0ms
(* (pow k m) a)
1.0ms
(* k (+ k 10))

series684.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

513.0ms
(/ (* (pow k m) a) (+ 1 (* k (+ k 10))))
151.0ms
(* (pow k m) a)
20.0ms
(* k (+ k 10))

simplify2.0s

Counts
32 → 56
Calls

32 calls. Slowest were:

540.0ms
(- (+ (* 99 (/ (* (exp (* -1 (* (log (/ 1 k)) m))) a) (pow k 4))) (/ (* (exp (* -1 (* (log (/ 1 k)) m))) a) (pow k 2))) (* 10 (/ (* (exp (* -1 (* (log (/ 1 k)) m))) a) (pow k 3))))
310.0ms
(- (+ (* 99 (/ (* a (exp (* m (- (log -1) (log (/ -1 k)))))) (pow k 4))) (/ (* a (exp (* m (- (log -1) (log (/ -1 k)))))) (pow k 2))) (* 10 (/ (* a (exp (* m (- (log -1) (log (/ -1 k)))))) (pow k 3))))
155.0ms
(+ (* 1 1) (- (* (* k (+ k 10)) (* k (+ k 10))) (* 1 (* k (+ k 10)))))

prune723.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 2.3b

localize15.0ms

Local error

Found 4 expressions with local error:

2.3b
(/ (* (sqrt (pow k m)) (* (sqrt (pow k m)) a)) (+ 1 (* k (+ k 10))))
0.0b
(* k (+ k 10))
0.0b
(* (sqrt (pow k m)) (* (sqrt (pow k m)) a))
0.0b
(* (sqrt (pow k m)) a)

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

15.0ms
(/ (* (sqrt (pow k m)) (* (sqrt (pow k m)) a)) (+ 1 (* k (+ k 10))))
8.0ms
(* (sqrt (pow k m)) (* (sqrt (pow k m)) a))
3.0ms
(* (sqrt (pow k m)) a)

series403.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

137.0ms
(* (sqrt (pow k m)) a)
123.0ms
(/ (* (sqrt (pow k m)) (* (sqrt (pow k m)) a)) (+ 1 (* k (+ k 10))))
111.0ms
(* (sqrt (pow k m)) (* (sqrt (pow k m)) a))
32.0ms
(* k (+ k 10))

simplify3.0s

Counts
48 → 82
Calls

48 calls. Slowest were:

556.0ms
(* (* (* (sqrt (pow k m)) (sqrt (pow k m))) (sqrt (pow k m))) (* (* (* (sqrt (pow k m)) a) (* (sqrt (pow k m)) a)) (* (sqrt (pow k m)) a)))
471.0ms
(- (+ (* 99 (/ (* (exp (* -1 (* (log (/ 1 k)) m))) a) (pow k 4))) (/ (* (exp (* -1 (* (log (/ 1 k)) m))) a) (pow k 2))) (* 10 (/ (* (exp (* -1 (* (log (/ 1 k)) m))) a) (pow k 3))))
241.0ms
(- (+ (* 99 (/ (* a (exp (* m (- (log -1) (log (/ -1 k)))))) (pow k 4))) (/ (* a (exp (* m (- (log -1) (log (/ -1 k)))))) (pow k 2))) (* 10 (/ (* a (exp (* m (- (log -1) (log (/ -1 k)))))) (pow k 3))))

prune1.3s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 2.3b

localize22.0ms

Local error

Found 4 expressions with local error:

14.0b
(sqrt (+ 1 (* k (+ k 10))))
14.0b
(sqrt (+ 1 (* k (+ k 10))))
0.0b
(* (/ (pow k m) (sqrt (+ 1 (* k (+ k 10))))) (/ a (sqrt (+ 1 (* k (+ k 10))))))
0.0b
(* k (+ k 10))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

6.0ms
(* (/ (pow k m) (sqrt (+ 1 (* k (+ k 10))))) (/ a (sqrt (+ 1 (* k (+ k 10))))))
1.0ms
(sqrt (+ 1 (* k (+ k 10))))
1.0ms
(sqrt (+ 1 (* k (+ k 10))))

series153.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

92.0ms
(* (/ (pow k m) (sqrt (+ 1 (* k (+ k 10))))) (/ a (sqrt (+ 1 (* k (+ k 10))))))
28.0ms
(* k (+ k 10))
17.0ms
(sqrt (+ 1 (* k (+ k 10))))
16.0ms
(sqrt (+ 1 (* k (+ k 10))))

simplify6.2s

Counts
70 → 102
Calls

70 calls. Slowest were:

666.0ms
(- (+ (* 99 (/ (* (exp (* -1 (* (log (/ 1 k)) m))) a) (pow k 4))) (/ (* (exp (* -1 (* (log (/ 1 k)) m))) a) (pow k 2))) (* 10 (/ (* (exp (* -1 (* (log (/ 1 k)) m))) a) (pow k 3))))
658.0ms
(/ (pow k m) (sqrt (+ (pow 1 3) (pow (* k (+ k 10)) 3))))
561.0ms
(sqrt (+ (pow 1 3) (pow (* k (+ k 10)) 3)))

prune1.6s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 2.3b

localize30.0ms

Local error

Found 4 expressions with local error:

14.4b
(cbrt (+ 1 (* (+ k 10) k)))
14.4b
(cbrt (+ 1 (* k (+ k 10))))
14.0b
(sqrt (+ 1 (* k (+ k 10))))
0.2b
(* (fabs (cbrt (+ 1 (* (+ k 10) k)))) (sqrt (cbrt (+ 1 (* k (+ k 10))))))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

9.0ms
(* (fabs (cbrt (+ 1 (* (+ k 10) k)))) (sqrt (cbrt (+ 1 (* k (+ k 10))))))
1.0ms
(cbrt (+ 1 (* (+ k 10) k)))
1.0ms
(sqrt (+ 1 (* k (+ k 10))))

series202.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

72.0ms
(* (fabs (cbrt (+ 1 (* (+ k 10) k)))) (sqrt (cbrt (+ 1 (* k (+ k 10))))))
61.0ms
(cbrt (+ 1 (* (+ k 10) k)))
42.0ms
(cbrt (+ 1 (* k (+ k 10))))
27.0ms
(sqrt (+ 1 (* k (+ k 10))))

simplify4.1s

Counts
44 → 77
Calls

44 calls. Slowest were:

581.0ms
(cbrt (+ (pow 1 3) (pow (* k (+ k 10)) 3)))
535.0ms
(sqrt (+ (pow 1 3) (pow (* k (+ k 10)) 3)))
281.0ms
(cbrt (- (* 1 1) (* (* k (+ k 10)) (* k (+ k 10)))))

prune1.7s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 2.3b

regimes183.0ms

Accuracy

0% (0.0b remaining)

Error of 2.0b against oracle of 2.0b and baseline of 2.0b

bsearch2.0ms