Average Error: 2.0 → 0.2
Time: 36.9s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{{k}^{m}}{\frac{1}{a} + \left(10 + k\right) \cdot \frac{k}{a}}\]

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 2.0

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

  2. Simplified2.0

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

  4. Applied associate-/l*2.1

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

  5. Taylor expanded around inf 4.3

    \[\leadsto \frac{{k}^{m}}{\color{blue}{\frac{1}{a} + \left(10 \cdot \frac{k}{a} + \frac{{k}^{2}}{a}\right)}}\]

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019005 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))

Details

Time bar (total: 36.4s)Debug log

sample162.0ms

Algorithm
intervals

simplify56.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

55.0ms
(/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k)))

prune39.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 1.7b

localize39.0ms

Local error

Found 2 expressions with local error:

1.7b
(/ (* (pow k m) a) (+ 1 (* k (+ k 10))))
0.0b
(* k (+ k 10))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
2 → 36
Calls

2 calls. Slowest were:

11.0ms
(/ (* (pow k m) a) (+ 1 (* k (+ k 10))))
2.0ms
(* k (+ k 10))

series150.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

118.0ms
(/ (* (pow k m) a) (+ 1 (* k (+ k 10))))
32.0ms
(* k (+ k 10))

simplify1.6s

Counts
27 → 42
Calls

27 calls. Slowest were:

531.0ms
(- (+ (* 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))))
209.0ms
(- (+ (* 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))))
147.0ms
(+ (* 1 1) (- (* (* k (+ k 10)) (* k (+ k 10))) (* 1 (* k (+ k 10)))))

prune468.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 1.6b

localize18.0ms

Local error

Found 4 expressions with local error:

13.5b
(sqrt (+ 1 (* k (+ k 10))))
13.5b
(sqrt (+ 1 (* k (+ k 10))))
0.0b
(/ (/ (* (pow k m) a) (sqrt (+ 1 (* k (+ k 10))))) (sqrt (+ 1 (* k (+ k 10)))))
0.0b
(* k (+ k 10))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 150
Calls

4 calls. Slowest were:

19.0ms
(/ (/ (* (pow k m) a) (sqrt (+ 1 (* k (+ k 10))))) (sqrt (+ 1 (* k (+ k 10)))))
2.0ms
(* k (+ k 10))
1.0ms
(sqrt (+ 1 (* k (+ k 10))))

series183.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

103.0ms
(/ (/ (* (pow k m) a) (sqrt (+ 1 (* k (+ k 10))))) (sqrt (+ 1 (* k (+ k 10)))))
31.0ms
(* k (+ k 10))
25.0ms
(sqrt (+ 1 (* k (+ k 10))))
24.0ms
(sqrt (+ 1 (* k (+ k 10))))

simplify15.4s

Counts
199 → 162
Calls

199 calls. Slowest were:

565.0ms
(/ (/ (* (pow k m) a) (sqrt (+ (pow 1 3) (pow (* k (+ k 10)) 3)))) 1)
564.0ms
(sqrt (+ (pow 1 3) (pow (* k (+ k 10)) 3)))
534.0ms
(/ (/ (* (pow k m) a) (sqrt (+ (pow 1 3) (pow (* k (+ k 10)) 3)))) (sqrt (* (cbrt (+ 1 (* k (+ k 10)))) (cbrt (+ 1 (* k (+ k 10)))))))

prune2.2s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 1.6b

localize15.0ms

Local error

Found 4 expressions with local error:

13.5b
(sqrt (+ 1 (* k (+ k 10))))
13.5b
(sqrt (+ 1 (* k (+ k 10))))
0.0b
(* (/ (pow k m) (sqrt (+ 1 (* k (+ k 10))))) (/ a (sqrt (+ 1 (* k (+ k 10))))))
0.0b
(/ (pow k m) (sqrt (+ 1 (* k (+ k 10)))))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 117
Calls

4 calls. Slowest were:

12.0ms
(* (/ (pow k m) (sqrt (+ 1 (* k (+ k 10))))) (/ a (sqrt (+ 1 (* k (+ k 10))))))
7.0ms
(/ (pow k m) (sqrt (+ 1 (* k (+ k 10)))))
1.0ms
(sqrt (+ 1 (* k (+ k 10))))

series254.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

130.0ms
(* (/ (pow k m) (sqrt (+ 1 (* k (+ k 10))))) (/ a (sqrt (+ 1 (* k (+ k 10))))))
79.0ms
(/ (pow k m) (sqrt (+ 1 (* k (+ k 10)))))
23.0ms
(sqrt (+ 1 (* k (+ k 10))))
23.0ms
(sqrt (+ 1 (* k (+ k 10))))

simplify6.8s

Counts
112 → 129
Calls

112 calls. Slowest were:

558.0ms
(* (* (* (/ (pow k m) (sqrt (+ 1 (* k (+ k 10))))) (/ (pow k m) (sqrt (+ 1 (* k (+ k 10)))))) (/ (pow k m) (sqrt (+ 1 (* k (+ k 10)))))) (* (* (/ a (sqrt (+ 1 (* k (+ k 10))))) (/ a (sqrt (+ 1 (* k (+ k 10)))))) (/ a (sqrt (+ 1 (* k (+ k 10)))))))
496.0ms
(sqrt (+ (pow 1 3) (pow (* k (+ k 10)) 3)))
470.0ms
(sqrt (+ (pow 1 3) (pow (* k (+ k 10)) 3)))

prune1.7s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 1.6b

localize6.0ms

Local error

Found 3 expressions with local error:

3.8b
(/ (+ 1 (* k (+ k 10))) a)
0.1b
(/ (pow k m) (/ (+ 1 (* k (+ k 10))) a))
0.0b
(* k (+ k 10))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
3 → 63
Calls

3 calls. Slowest were:

6.0ms
(/ (+ 1 (* k (+ k 10))) a)
4.0ms
(/ (pow k m) (/ (+ 1 (* k (+ k 10))) a))
1.0ms
(* k (+ k 10))

series114.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

77.0ms
(/ (pow k m) (/ (+ 1 (* k (+ k 10))) a))
19.0ms
(* k (+ k 10))
18.0ms
(/ (+ 1 (* k (+ k 10))) a)

simplify3.3s

Counts
53 → 72
Calls

53 calls. Slowest were:

736.0ms
(/ (* (* (pow k m) (pow k m)) (pow k m)) (* (* (/ (+ 1 (* k (+ k 10))) a) (/ (+ 1 (* k (+ k 10))) a)) (/ (+ 1 (* k (+ k 10))) a)))
449.0ms
(- (+ (* 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))))
384.0ms
(+ (pow 1 3) (pow (* k (+ k 10)) 3))

prune674.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

regimes115.0ms

Accuracy

0% (0.1b remaining)

Error of 0.2b against oracle of 0.1b and baseline of 0.2b

bsearch2.0ms

end0.0ms

sample3.1s

Algorithm
intervals