Average Error: 30.0 → 0.6
Time: 48.1s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(\sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x + 1}\right) + \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\]

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

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

  7. Taylor expanded around 0 1.8

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

  8. Simplified0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

Time bar (total: 46.6s)Debug log

sample73.0ms

Algorithm
intervals

simplify4.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
4.0ms
(- (cbrt (+ x 1)) (cbrt x))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 25.8b

localize20.0ms

Local error

Found 3 expressions with local error:

3.0b
(- (cbrt (+ x 1)) (cbrt x))
0.6b
(cbrt x)
0.2b
(cbrt (+ x 1))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
22×add-sqr-sqrt
20×*-un-lft-identity
18×add-cube-cbrt
18×prod-diff
16×cbrt-prod
6×fma-neg
5×add-log-exp
3×log1p-expm1-u
3×add-exp-log
3×add-cbrt-cube
3×pow1
3×expm1-log1p-u
2×difference-of-squares
2×pow1/3
2×cbrt-div
1×distribute-lft-out--
1×flip--
1×flip-+
1×diff-log
1×flip3--
1×flip3-+
1×sub-neg
Counts
3 → 66
Calls
3 calls:
Slowest
7.0ms
(- (cbrt (+ x 1)) (cbrt x))
1.0ms
(cbrt (+ x 1))
0.0ms
(cbrt x)

series451.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
294.0ms
(cbrt x)
124.0ms
(- (cbrt (+ x 1)) (cbrt x))
32.0ms
(cbrt (+ x 1))

simplify1.8s

Counts
61 → 75
Calls
61 calls:
Slowest
295.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
293.0ms
(- (+ (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)))))
206.0ms
(- (+ (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))))
89.0ms
(- (+ (* 1/3 x) 1) (* 1/9 (pow x 2)))
64.0ms
(- (+ (* 1/3 (pow (/ 1 (pow x 2)) 1/3)) (* 5/81 (pow (/ 1 (pow x 8)) 1/3))) (* 1/9 (pow (/ 1 (pow x 5)) 1/3)))

prune648.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 11.4b

localize31.0ms

Local error

Found 4 expressions with local error:

2.7b
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))
0.7b
(pow (cbrt x) 3)
0.6b
(cbrt x)
0.6b
(cbrt x)

rewrite28.0ms

Algorithm
rewrite-expression-head
Rules
187×prod-diff
147×add-sqr-sqrt
141×cube-prod
141×unpow-prod-down
127×*-un-lft-identity
117×add-cube-cbrt
78×cbrt-prod
30×unpow3
30×cube-mult
17×fma-neg
15×difference-of-squares
6×add-log-exp
5×distribute-lft-out--
5×add-exp-log
5×pow1
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
3×pow1/3
2×pow-pow
1×rem-cube-cbrt
1×flip--
1×pow-exp
1×diff-log
1×flip3--
1×sub-neg
1×difference-cubes
1×pow-to-exp
Counts
4 → 281
Calls
4 calls:
Slowest
17.0ms
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))
1.0ms
(pow (cbrt x) 3)
0.0ms
(cbrt x)
0.0ms
(cbrt x)

series1000.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
370.0ms
(pow (cbrt x) 3)
300.0ms
(cbrt x)
272.0ms
(cbrt x)
58.0ms
(- (pow (cbrt (+ x 1)) 3) (pow (cbrt x) 3))

simplify33.3s

Counts
452 → 293
Calls
452 calls:
Slowest
587.0ms
(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)))
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)))
335.0ms
(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)))
325.0ms
(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)))
323.0ms
(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)))

prune2.9s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.3b

localize8.0ms

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
Rules
9×pow1
7×pow1/3
6×add-cube-cbrt
6×add-exp-log
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
3×associate-*l*
3×associate-*r*
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 54
Calls
4 calls:
Slowest
4.0ms
(* (cbrt x) (cbrt x))
0.0ms
(cbrt x)
0.0ms
(cbrt x)
0.0ms
(cbrt x)

series1.1s

Counts
4 → 12
Calls
4 calls:
Slowest
278.0ms
(cbrt x)
272.0ms
(cbrt x)
259.0ms
(* (cbrt x) (cbrt x))
249.0ms
(cbrt x)

simplify77.0ms

Counts
25 → 66
Calls
25 calls:
Slowest
12.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
10.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
9.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
9.0ms
(* (pow (pow x 2) 1/3) (pow (cbrt -1) 2))
4.0ms
(pow (/ 1 x) -1/3)

prune639.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.2b

localize26.0ms

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
Rules
4×add-log-exp
4×log1p-expm1-u
4×add-cube-cbrt
4×add-exp-log
4×pow1/3
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×expm1-log1p-u
4×add-sqr-sqrt
Counts
4 → 40
Calls
4 calls:
Slowest
0.0ms
(cbrt x)
0.0ms
(cbrt x)
0.0ms
(cbrt x)
0.0ms
(cbrt x)

series1.1s

Counts
4 → 12
Calls
4 calls:
Slowest
295.0ms
(cbrt x)
289.0ms
(cbrt x)
274.0ms
(cbrt x)
270.0ms
(cbrt x)

simplify55.0ms

Counts
12 → 52
Calls
12 calls:
Slowest
9.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
8.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
8.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
8.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
3.0ms
(pow (/ 1 x) -1/3)

prune618.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.2b

regimes201.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample2.7s

Algorithm
intervals