Average Error: 30.0 → 0.6
Time: 47.8s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\left(\sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}\right) \cdot \left(\sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.0

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

  3. Applied flip3--30.0

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

  4. Taylor expanded around inf 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. Using strategy rm

  8. Applied add-sqr-sqrt0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

Time bar (total: 46.6s)Debug log

sample51.0ms

Algorithm
intervals

simplify4.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

4.0ms
(- (cbrt (+ x 1)) (cbrt x))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.3b

localize24.0ms

Local error

Found 3 expressions with local error:

2.8b
(- (cbrt (+ x 1)) (cbrt x))
0.6b
(cbrt x)
0.2b
(cbrt (+ x 1))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 66
Calls

3 calls. Slowest were:

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

series452.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

328.0ms
(cbrt x)
87.0ms
(- (cbrt (+ x 1)) (cbrt x))
36.0ms
(cbrt (+ x 1))

simplify2.1s

Counts
61 → 75
Calls

61 calls. Slowest were:

395.0ms
(- (+ (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)))))
283.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
267.0ms
(- (+ (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))))

prune642.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 15.5b

localize32.0ms

Local error

Found 4 expressions with local error:

1.7b
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))
0.7b
(pow (cbrt x) 3)
0.6b
(cbrt x)
0.6b
(cbrt x)

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 281
Calls

4 calls. Slowest were:

13.0ms
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))
1.0ms
(pow (cbrt x) 3)
0.0ms
(cbrt x)

series1.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

343.0ms
(pow (cbrt x) 3)
323.0ms
(cbrt x)
288.0ms
(cbrt x)
85.0ms
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))

simplify33.1s

Counts
452 → 293
Calls

452 calls. Slowest were:

370.0ms
(fma (- (pow (cbrt (cbrt x)) 3)) (pow (* (cbrt (cbrt x)) (cbrt (cbrt x))) 3) (* (pow (cbrt (cbrt x)) 3) (pow (* (cbrt (cbrt x)) (cbrt (cbrt x))) 3)))
351.0ms
(fma (- (pow (cbrt (cbrt x)) 3)) (pow (* (cbrt (cbrt x)) (cbrt (cbrt x))) 3) (* (pow (cbrt (cbrt x)) 3) (pow (* (cbrt (cbrt x)) (cbrt (cbrt x))) 3)))
342.0ms
(fma (- (pow (cbrt (cbrt x)) 3)) (pow (* (cbrt (cbrt x)) (cbrt (cbrt x))) 3) (* (pow (cbrt (cbrt x)) 3) (pow (* (cbrt (cbrt x)) (cbrt (cbrt x))) 3)))

prune3.3s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.4b

localize5.0ms

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))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
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:

318.0ms
(cbrt x)
286.0ms
(* (cbrt x) (cbrt x))
267.0ms
(cbrt x)
264.0ms
(cbrt x)

simplify87.0ms

Counts
25 → 66
Calls

25 calls. Slowest were:

19.0ms
(* (pow (pow x 2) 1/3) (pow (cbrt -1) 2))
10.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
10.0ms
(* (pow (* x -1) 1/3) (cbrt -1))

prune675.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.3b

localize35.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt x)
0.6b
(cbrt x)
0.6b
(cbrt x)
0.6b
(cbrt x)

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
4 → 40
Calls

4 calls. Slowest were:

0.0ms
(cbrt x)
0.0ms
(cbrt x)
0.0ms
(cbrt x)

series1.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

301.0ms
(cbrt x)
293.0ms
(cbrt x)
275.0ms
(cbrt x)
270.0ms
(cbrt x)

simplify35.0ms

Counts
12 → 52
Calls

12 calls. Slowest were:

8.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
5.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
5.0ms
(* (pow (* x -1) 1/3) (cbrt -1))

prune627.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.2b

regimes61.0ms

Accuracy

0% (0.3b remaining)

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

bsearch2.0ms

end0.0ms

sample2.1s

Algorithm
intervals