Average Error: 29.7 → 0.6
Time: 13.2s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\left(\sqrt[3]{x} \cdot \sqrt{\sqrt[3]{x}}\right) \cdot \sqrt{\sqrt[3]{x}} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.7

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

  3. Applied flip3--29.7

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

  4. Taylor expanded around 0 0.6

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

  6. Applied add-sqr-sqrt0.6

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

  7. Applied associate-*l*0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2018362 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

Time bar (total: 9.5s)Debug log

start96.0ms

Algorithm
intervals

setup34.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 27.8b

localize10.0ms

Local error

Found 3 expressions with local error:

1.6b
(- (cbrt (+ x 1)) (cbrt x))
0.6b
(cbrt x)
0.2b
(cbrt (+ x 1))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
3 → 36
Calls

3 calls. Slowest were:

5.0ms
(- (cbrt (+ x 1)) (cbrt x))
0.0ms
(cbrt (+ x 1))
0.0ms
(cbrt x)

series515.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

361.0ms
(cbrt x)
118.0ms
(- (cbrt (+ x 1)) (cbrt x))
36.0ms
(cbrt (+ x 1))

simplify917.0ms

Counts
20 → 45
Calls

20 calls. Slowest were:

260.0ms
(- (+ (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)))))
233.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
183.0ms
(- (+ (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))))

prune520.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 14.5b

localize22.0ms

Local error

Found 3 expressions with local error:

28.1b
(- (cbrt (+ x 1)) (pow x 1/3))
5.2b
(pow x 1/3)
0.2b
(cbrt (+ x 1))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
3 → 37
Calls

3 calls. Slowest were:

11.0ms
(- (cbrt (+ x 1)) (pow x 1/3))
1.0ms
(cbrt (+ x 1))
0.0ms
(pow x 1/3)

series378.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

214.0ms
(pow x 1/3)
109.0ms
(- (cbrt (+ x 1)) (pow x 1/3))
55.0ms
(cbrt (+ x 1))

simplify1.2s

Counts
20 → 46
Calls

20 calls. Slowest were:

306.0ms
(- (+ (* 5/81 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (pow x 3))) (* 1/3 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) x))) (* 1/9 (/ (exp (* 1/3 (- (log -1) (log (/ -1 x))))) (pow x 2))))
226.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
207.0ms
(- (+ (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))))

prune544.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.3b

localize37.0ms

Local error

Found 4 expressions with local error:

1.5b
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))
0.7b
(pow (cbrt x) 3)
0.6b
(cbrt x)
0.6b
(cbrt x)

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

21.0ms
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))
2.0ms
(pow (cbrt x) 3)
0.0ms
(cbrt x)

series989.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

333.0ms
(cbrt x)
296.0ms
(pow (cbrt x) 3)
283.0ms
(cbrt x)
77.0ms
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))

simplify527.0ms

Counts
62 → 81
Calls

62 calls. Slowest were:

66.0ms
(/ (exp (pow (cbrt (+ x 1)) 3)) (exp (pow (cbrt x) 3)))
23.0ms
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))
20.0ms
(+ (* (cbrt (+ x 1)) (cbrt (+ x 1))) (+ (* (cbrt x) (cbrt x)) (* (cbrt (+ x 1)) (cbrt x))))

prune1.5s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.2b

localize9.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt x)
0.6b
(cbrt x)
0.6b
(cbrt x)
0.5b
(* (cbrt x) (cbrt x))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 46
Calls

4 calls. Slowest were:

6.0ms
(* (cbrt x) (cbrt x))
0.0ms
(cbrt x)
0.0ms
(cbrt x)

series1.2s

Counts
4 → 12
Calls

4 calls. Slowest were:

343.0ms
(* (cbrt x) (cbrt x))
328.0ms
(cbrt x)
292.0ms
(cbrt x)
286.0ms
(cbrt x)

simplify67.0ms

Counts
25 → 58
Calls

25 calls. Slowest were:

17.0ms
(* (pow (pow x 2) 1/3) (pow (cbrt -1) 2))
6.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
5.0ms
(* (pow (* x -1) 1/3) (cbrt -1))

prune688.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.2b

regimes115.0ms

Accuracy

0% (0.4b remaining)

Error of 0.6b against oracle of 0.2b and baseline of 0.6b

bsearch4.0ms