Average Error: 30.1 → 0.6
Time: 15.2s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)} \cdot \sqrt[3]{x} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \sqrt[3]{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.1

    \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube30.0

    \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} - \sqrt[3]{x}\]
  4. Using strategy rm

  5. Applied flip3--30.0

    \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{x}\right)}}\]

  6. Taylor expanded around 0 0.6

    \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{x}\right)}\]

  7. Final simplification0.6

    \[\leadsto \frac{1}{\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)} \cdot \sqrt[3]{x} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \sqrt[3]{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

Time bar (total: 14.1s)Debug log

sample94.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

8.0ms
(- (cbrt (+ x 1)) (cbrt x))

prune25.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 32.3b

localize15.0ms

Local error

Found 3 expressions with local error:

3.3b
(- (cbrt (+ x 1)) (cbrt x))
0.5b
(cbrt x)
0.3b
(cbrt (+ x 1))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
3 → 36
Calls

3 calls. Slowest were:

6.0ms
(- (cbrt (+ x 1)) (cbrt x))
1.0ms
(cbrt (+ x 1))
0.0ms
(cbrt x)

series523.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

365.0ms
(cbrt x)
107.0ms
(- (cbrt (+ x 1)) (cbrt x))
51.0ms
(cbrt (+ x 1))

simplify828.0ms

Counts
20 → 45
Calls

20 calls. Slowest were:

222.0ms
(- (+ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (* 1/3 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) x))) (+ (* (pow (* x -1) 1/3) (cbrt -1)) (* 1/9 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (pow x 2)))))
213.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
155.0ms
(- (+ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (* 1/3 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) x))) (* 1/9 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (pow x 2))))

prune425.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 16.4b

localize24.0ms

Local error

Found 4 expressions with local error:

3.3b
(- (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1)))) (cbrt x))
0.5b
(cbrt x)
0.3b
(* (cbrt (+ x 1)) (cbrt (+ x 1)))
0.3b
(cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1))))

rewrite53.0ms

Algorithm
rewrite-expression-head
Counts
4 → 94
Calls

4 calls. Slowest were:

28.0ms
(- (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1)))) (cbrt x))
12.0ms
(* (cbrt (+ x 1)) (cbrt (+ x 1)))
10.0ms
(cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1))))

series442.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

274.0ms
(cbrt x)
92.0ms
(- (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1)))) (cbrt x))
39.0ms
(cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1))))
37.0ms
(* (cbrt (+ x 1)) (cbrt (+ x 1)))

simplify2.8s

Counts
76 → 106
Calls

76 calls. Slowest were:

328.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
249.0ms
(- (+ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (* 1/3 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) x))) (+ (* (pow (* x -1) 1/3) (cbrt -1)) (* 1/9 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (pow x 2)))))
155.0ms
(- (+ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (* 1/3 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) x))) (* 1/9 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (pow x 2))))

prune1.2s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 16.4b

localize63.0ms

Local error

Found 4 expressions with local error:

2.7b
(- (pow (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1)))) 3) (pow (cbrt x) 3))
0.7b
(pow (cbrt x) 3)
0.5b
(cbrt x)
0.5b
(cbrt x)

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 63
Calls

4 calls. Slowest were:

20.0ms
(- (pow (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1)))) 3) (pow (cbrt x) 3))
2.0ms
(pow (cbrt x) 3)
0.0ms
(cbrt x)

series985.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

326.0ms
(cbrt x)
303.0ms
(cbrt x)
273.0ms
(pow (cbrt x) 3)
82.0ms
(- (pow (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1)))) 3) (pow (cbrt x) 3))

simplify753.0ms

Counts
50 → 75
Calls

50 calls. Slowest were:

61.0ms
(- (pow (sqrt (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1))))) 3) (pow (sqrt (cbrt x)) 3))
50.0ms
(+ (pow (sqrt (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1))))) 3) (pow (sqrt (cbrt x)) 3))
44.0ms
(- (pow (sqrt (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1))))) 3) (pow (sqrt (cbrt x)) 3))

prune1.1s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.4b

localize13.0ms

Local error

Found 4 expressions with local error:

0.5b
(cbrt x)
0.5b
(cbrt x)
0.5b
(cbrt x)
0.5b
(* (cbrt x) (cbrt x))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 46
Calls

4 calls. Slowest were:

4.0ms
(* (cbrt x) (cbrt x))
0.0ms
(cbrt x)
0.0ms
(cbrt x)

series1.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

333.0ms
(* (cbrt x) (cbrt x))
279.0ms
(cbrt x)
272.0ms
(cbrt x)
260.0ms
(cbrt x)

simplify91.0ms

Counts
25 → 58
Calls

25 calls. Slowest were:

18.0ms
(* (pow (pow x 2) 1/3) (pow (cbrt -1) 2))
11.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
10.0ms
(* (pow (* x -1) 1/3) (cbrt -1))

prune668.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.3b

regimes82.0ms

Accuracy

0% (0.3b remaining)

Error of 0.6b against oracle of 0.3b and baseline of 0.6b

bsearch4.0ms

end0.0ms

sample2.7s

Algorithm
intervals