Average Error: 29.7 → 0.6
Time: 51.2s
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 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

Time bar (total: 47.4s)Debug log

start55.0ms

Algorithm
intervals

setup24.0ms

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))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 66
Calls

3 calls. Slowest were:

7.0ms
(- (cbrt (+ x 1)) (cbrt x))
1.0ms
(cbrt (+ x 1))
0.0ms
(cbrt x)

series452.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

308.0ms
(cbrt x)
97.0ms
(- (cbrt (+ x 1)) (cbrt x))
46.0ms
(cbrt (+ x 1))

simplify2.0s

Counts
61 → 75
Calls

61 calls. Slowest were:

440.0ms
(- (+ (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)))))
385.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
252.0ms
(- (+ (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))))

prune747.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 14.5b

localize18.0ms

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))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
3 → 67
Calls

3 calls. Slowest were:

16.0ms
(- (cbrt (+ x 1)) (pow x 1/3))
1.0ms
(cbrt (+ x 1))
0.0ms
(pow x 1/3)

series486.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

326.0ms
(pow x 1/3)
107.0ms
(- (cbrt (+ x 1)) (pow x 1/3))
54.0ms
(cbrt (+ x 1))

simplify2.5s

Counts
61 → 76
Calls

61 calls. Slowest were:

412.0ms
(- (+ (* 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))))
389.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
179.0ms
(- (+ (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))))

prune791.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.3b

localize41.0ms

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)

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 281
Calls

4 calls. Slowest were:

14.0ms
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))
1.0ms
(pow (cbrt x) 3)
0.0ms
(cbrt x)

series1.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

346.0ms
(cbrt x)
328.0ms
(cbrt x)
311.0ms
(pow (cbrt x) 3)
80.0ms
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))

simplify33.5s

Counts
452 → 293
Calls

452 calls. Slowest were:

411.0ms
(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)))
384.0ms
(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)))
373.0ms
(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)))

prune3.5s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.2b

localize6.0ms

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)

series1.2s

Counts
4 → 12
Calls

4 calls. Slowest were:

332.0ms
(cbrt x)
317.0ms
(* (cbrt x) (cbrt x))
306.0ms
(cbrt x)
263.0ms
(cbrt x)

simplify81.0ms

Counts
25 → 66
Calls

25 calls. Slowest were:

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

prune742.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.2b

regimes71.0ms

Accuracy

0% (0.4b remaining)

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

bsearch2.0ms