Average Error: 1.9 → 1.8
Time: 22.7s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{1}{1 + \left(k + 10\right) \cdot k} \cdot \left(\sqrt{{k}^{m}} \cdot \left(\sqrt{{k}^{m}} \cdot a\right)\right)\]

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 1.9

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

  2. Simplified1.8

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

  4. Applied add-sqr-sqrt1.8

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

  5. Applied associate-*l*1.8

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

  7. Applied div-inv1.8

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

  8. Final simplification1.8

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

Reproduce

herbie shell --seed 2019007 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))

Details

Time bar (total: 21.7s)Debug log

sample129.0ms

Algorithm
intervals

simplify54.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

54.0ms
(/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.2b

localize17.0ms

Local error

Found 3 expressions with local error:

1.2b
(/ (* (pow k m) a) (+ 1 (* k (+ k 10))))
0.0b
(* (pow k m) a)
0.0b
(* k (+ k 10))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
3 → 47
Calls

3 calls. Slowest were:

8.0ms
(/ (* (pow k m) a) (+ 1 (* k (+ k 10))))
4.0ms
(* (pow k m) a)
2.0ms
(* k (+ k 10))

series277.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

141.0ms
(* (pow k m) a)
105.0ms
(/ (* (pow k m) a) (+ 1 (* k (+ k 10))))
31.0ms
(* k (+ k 10))

simplify1.9s

Counts
32 → 56
Calls

32 calls. Slowest were:

464.0ms
(- (+ (* 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))))
312.0ms
(- (+ (* 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))))
175.0ms
(/ (* (* (* (pow k m) a) (* (pow k m) a)) (* (pow k m) a)) (* (* (+ 1 (* k (+ k 10))) (+ 1 (* k (+ k 10)))) (+ 1 (* k (+ k 10)))))

prune629.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 1.1b

localize13.0ms

Local error

Found 4 expressions with local error:

1.2b
(/ (* (sqrt (pow k m)) (* (sqrt (pow k m)) a)) (+ 1 (* k (+ k 10))))
0.0b
(* (sqrt (pow k m)) a)
0.0b
(* k (+ k 10))
0.0b
(* (sqrt (pow k m)) (* (sqrt (pow k m)) a))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 70
Calls

4 calls. Slowest were:

12.0ms
(* (sqrt (pow k m)) (* (sqrt (pow k m)) a))
11.0ms
(/ (* (sqrt (pow k m)) (* (sqrt (pow k m)) a)) (+ 1 (* k (+ k 10))))
3.0ms
(* (sqrt (pow k m)) a)

series392.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

153.0ms
(* (sqrt (pow k m)) a)
105.0ms
(* (sqrt (pow k m)) (* (sqrt (pow k m)) a))
100.0ms
(/ (* (sqrt (pow k m)) (* (sqrt (pow k m)) a)) (+ 1 (* k (+ k 10))))
33.0ms
(* k (+ k 10))

simplify3.3s

Counts
48 → 82
Calls

48 calls. Slowest were:

525.0ms
(* (* (* (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)))
519.0ms
(- (+ (* 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))))
272.0ms
(/ (* (* (* (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))) (* (* (+ 1 (* k (+ k 10))) (+ 1 (* k (+ k 10)))) (+ 1 (* k (+ k 10)))))

prune1.1s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 1.1b

localize26.0ms

Local error

Found 4 expressions with local error:

1.2b
(/ (* (* (cbrt (pow k m)) (cbrt (pow k m))) (* (cbrt (pow k m)) a)) (+ 1 (* k (+ k 10))))
0.0b
(cbrt (pow k m))
0.0b
(cbrt (pow k m))
0.0b
(cbrt (pow k m))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 58
Calls

4 calls. Slowest were:

39.0ms
(/ (* (* (cbrt (pow k m)) (cbrt (pow k m))) (* (cbrt (pow k m)) a)) (+ 1 (* k (+ k 10))))
1.0ms
(cbrt (pow k m))
1.0ms
(cbrt (pow k m))

series457.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

126.0ms
(cbrt (pow k m))
114.0ms
(/ (* (* (cbrt (pow k m)) (cbrt (pow k m))) (* (cbrt (pow k m)) a)) (+ 1 (* k (+ k 10))))
111.0ms
(cbrt (pow k m))
106.0ms
(cbrt (pow k m))

simplify4.3s

Counts
39 → 70
Calls

39 calls. Slowest were:

550.0ms
(- (+ (* 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))))
392.0ms
(exp (* 1/3 (* m (- (log -1) (log (/ -1 k))))))
308.0ms
(exp (* 1/3 (* m (- (log -1) (log (/ -1 k))))))

prune964.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 1.1b

localize6.0ms

Local error

Found 4 expressions with local error:

1.2b
(* (* (sqrt (pow k m)) (* (sqrt (pow k m)) a)) (/ 1 (+ 1 (* k (+ k 10)))))
0.3b
(/ 1 (+ 1 (* k (+ k 10))))
0.0b
(* (sqrt (pow k m)) a)
0.0b
(* k (+ k 10))

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

20.0ms
(* (* (sqrt (pow k m)) (* (sqrt (pow k m)) a)) (/ 1 (+ 1 (* k (+ k 10)))))
4.0ms
(/ 1 (+ 1 (* k (+ k 10))))
3.0ms
(* (sqrt (pow k m)) a)

series275.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

156.0ms
(* (sqrt (pow k m)) a)
87.0ms
(* (* (sqrt (pow k m)) (* (sqrt (pow k m)) a)) (/ 1 (+ 1 (* k (+ k 10)))))
20.0ms
(* k (+ k 10))
11.0ms
(/ 1 (+ 1 (* k (+ k 10))))

simplify3.9s

Counts
47 → 84
Calls

47 calls. Slowest were:

487.0ms
(- (+ (/ 1 (pow k 2)) (* 99 (/ 1 (pow k 4)))) (* 10 (/ 1 (pow k 3))))
472.0ms
(- (+ (* 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))))
359.0ms
(+ (+ (log (sqrt (pow k m))) (log (* (sqrt (pow k m)) a))) (log (/ 1 (+ 1 (* k (+ k 10))))))

prune943.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 1.1b

regimes148.0ms

Accuracy

0% (0.0b remaining)

Error of 1.8b against oracle of 1.8b and baseline of 1.8b

bsearch2.0ms

end0.0ms

sample2.8s

Algorithm
intervals