Average Error: 29.6 → 0.6
Time: 48.6s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x + 1} + \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}\right)}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.6

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

  3. Applied flip3--29.5

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

  4. Taylor expanded around 0 0.5

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

  6. Applied add-cbrt-cube0.6

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

  8. Applied add-cbrt-cube0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

Time bar (total: 47.0s)Debug log

sample107.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 31.9b

localize22.0ms

Local error

Found 3 expressions with local error:

3.2b
(- (cbrt (+ x 1)) (cbrt x))
0.5b
(cbrt x)
0.3b
(cbrt (+ x 1))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
3 → 66
Calls

3 calls. Slowest were:

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

series416.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

261.0ms
(cbrt x)
117.0ms
(- (cbrt (+ x 1)) (cbrt x))
39.0ms
(cbrt (+ x 1))

simplify2.0s

Counts
61 → 75
Calls

61 calls. Slowest were:

373.0ms
(- (+ (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)))))
346.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
264.0ms
(- (+ (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))))

prune672.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 17.3b

localize29.0ms

Local error

Found 4 expressions with local error:

2.9b
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))
0.7b
(pow (cbrt x) 3)
0.5b
(cbrt x)
0.5b
(cbrt x)

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
4 → 281
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)

series1.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

336.0ms
(cbrt x)
312.0ms
(cbrt x)
308.0ms
(pow (cbrt x) 3)
83.0ms
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))

simplify33.1s

Counts
452 → 293
Calls

452 calls. Slowest were:

339.0ms
(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)))
332.0ms
(fma (pow (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) 3) (pow (cbrt (cbrt (+ x 1))) 3) (- (* (pow (cbrt (cbrt x)) 3) (pow (* (cbrt (cbrt x)) (cbrt (cbrt x))) 3))))
331.0ms
(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)))

prune3.0s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.4b

localize10.0ms

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))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

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

series1.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

310.0ms
(* (cbrt x) (cbrt x))
284.0ms
(cbrt x)
279.0ms
(cbrt x)
274.0ms
(cbrt x)

simplify70.0ms

Counts
25 → 66
Calls

25 calls. Slowest were:

11.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
9.0ms
(* (pow (pow x 2) 1/3) (pow (cbrt -1) 2))
8.0ms
(* (pow (* x -1) 1/3) (cbrt -1))

prune777.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 0.3b

localize21.0ms

Local error

Found 4 expressions with local error:

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

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)
297.0ms
(cbrt x)
282.0ms
(cbrt x)
245.0ms
(cbrt x)

simplify37.0ms

Counts
12 → 52
Calls

12 calls. Slowest were:

9.0ms
(* (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))

prune654.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 0.3b

regimes120.0ms

Accuracy

0% (0.3b remaining)

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

bsearch2.0ms

end0.0ms

sample2.7s

Algorithm
intervals