Average Error: 1.9 → 1.8
Time: 25.6s
Precision: 64
Internal Precision: 128
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\frac{\left(\sqrt[3]{{k}^{m}} \cdot a\right) \cdot \left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right)}{(k \cdot \left(k + 10\right) + 1)_*}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Initial program 1.9

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

  2. Simplified1.8

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

  4. Applied add-cube-cbrt1.8

    \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right) \cdot \sqrt[3]{{k}^{m}}\right)} \cdot a}{(k \cdot \left(k + 10\right) + 1)_*}\]

  5. Applied associate-*l*1.8

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right) \cdot \left(\sqrt[3]{{k}^{m}} \cdot a\right)}}{(k \cdot \left(k + 10\right) + 1)_*}\]

  6. Final simplification1.8

    \[\leadsto \frac{\left(\sqrt[3]{{k}^{m}} \cdot a\right) \cdot \left(\sqrt[3]{{k}^{m}} \cdot \sqrt[3]{{k}^{m}}\right)}{(k \cdot \left(k + 10\right) + 1)_*}\]

Reproduce

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

Details

Time bar (total: 24.7s)Debug log

sample87.0ms

Algorithm
intervals

simplify38.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

37.0ms
(/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k)))

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 1.2b

localize28.0ms

Local error

Found 2 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 34
Calls

2 calls. Slowest were:

6.0ms
(/ (* (pow k m) a) (fma k (+ k 10) 1))
2.0ms
(* (pow k m) a)

series239.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

121.0ms
(/ (* (pow k m) a) (fma k (+ k 10) 1))
118.0ms
(* (pow k m) a)

simplify2.8s

Counts
20 → 40
Calls

20 calls. Slowest were:

931.0ms
(/ (* (* (* (pow k m) a) (* (pow k m) a)) (* (pow k m) a)) (* (* (fma k (+ k 10) 1) (fma k (+ k 10) 1)) (fma k (+ k 10) 1)))
577.0ms
(- (+ (* 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))))
313.0ms
(- (+ (* 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))))

prune492.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.2b

localize10.0ms

Local error

Found 4 expressions with local error:

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

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 73
Calls

4 calls. Slowest were:

19.0ms
(/ (* (sqrt (pow k m)) (* (sqrt (pow k m)) a)) (fma k (+ k 10) 1))
18.0ms
(* (sqrt (pow k m)) (* (sqrt (pow k m)) a))
6.0ms
(* (sqrt (pow k m)) a)

series520.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

152.0ms
(* (sqrt (pow k m)) a)
142.0ms
(* (sqrt (pow k m)) (* (sqrt (pow k m)) a))
126.0ms
(/ (* (sqrt (pow k m)) (* (sqrt (pow k m)) a)) (fma k (+ k 10) 1))
100.0ms
(sqrt (pow k m))

simplify3.3s

Counts
43 → 85
Calls

43 calls. Slowest were:

628.0ms
(- (+ (* 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))))
475.0ms
(* (* (* (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)))
320.0ms
(- (+ (* 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))))

prune962.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.2b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

17.0ms
(/ (* (* (cbrt (pow k m)) (cbrt (pow k m))) (* (cbrt (pow k m)) a)) (fma k (+ k 10) 1))
1.0ms
(cbrt (pow k m))
1.0ms
(cbrt (pow k m))

series401.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

116.0ms
(cbrt (pow k m))
105.0ms
(cbrt (pow k m))
93.0ms
(/ (* (* (cbrt (pow k m)) (cbrt (pow k m))) (* (cbrt (pow k m)) a)) (fma k (+ k 10) 1))
87.0ms
(cbrt (pow k m))

simplify4.3s

Counts
37 → 76
Calls

37 calls. Slowest were:

486.0ms
(- (+ (* 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))))
475.0ms
(exp (* 1/3 (* m (- (log -1) (log (/ -1 k))))))
428.0ms
(exp (* 1/3 (* m (- (log -1) (log (/ -1 k))))))

prune996.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.2b

localize13.0ms

Local error

Found 4 expressions with local error:

15.4b
(cbrt (fma k (+ k 10) 1))
15.4b
(cbrt (fma k (+ k 10) 1))
15.4b
(cbrt (fma k (+ k 10) 1))
0.5b
(* (/ (sqrt (pow k m)) (* (cbrt (fma k (+ k 10) 1)) (cbrt (fma k (+ k 10) 1)))) (/ (* (sqrt (pow k m)) a) (cbrt (fma k (+ k 10) 1))))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

20.0ms
(* (/ (sqrt (pow k m)) (* (cbrt (fma k (+ k 10) 1)) (cbrt (fma k (+ k 10) 1)))) (/ (* (sqrt (pow k m)) a) (cbrt (fma k (+ k 10) 1))))
1.0ms
(cbrt (fma k (+ k 10) 1))
1.0ms
(cbrt (fma k (+ k 10) 1))

series275.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

91.0ms
(* (/ (sqrt (pow k m)) (* (cbrt (fma k (+ k 10) 1)) (cbrt (fma k (+ k 10) 1)))) (/ (* (sqrt (pow k m)) a) (cbrt (fma k (+ k 10) 1))))
64.0ms
(cbrt (fma k (+ k 10) 1))
61.0ms
(cbrt (fma k (+ k 10) 1))
59.0ms
(cbrt (fma k (+ k 10) 1))

simplify5.2s

Counts
57 → 97
Calls

57 calls. Slowest were:

891.0ms
(- (+ (* 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))))
324.0ms
(+ (log (/ (sqrt (pow k m)) (* (cbrt (fma k (+ k 10) 1)) (cbrt (fma k (+ k 10) 1))))) (- (log (* (sqrt (pow k m)) a)) (log (cbrt (fma k (+ k 10) 1)))))
291.0ms
(* (/ (sqrt (pow k m)) (* (cbrt (fma k (+ k 10) 1)) (cbrt (fma k (+ k 10) 1)))) (/ (* (sqrt (pow k m)) a) (cbrt (fma k (+ k 10) 1))))

prune1.3s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.2b

regimes229.0ms

Accuracy

0% (1.7b remaining)

Error of 1.8b against oracle of 0.1b and baseline of 1.8b

bsearch3.0ms

end0.0ms

sample3.5s

Algorithm
intervals