Average Error: 30.0 → 0.7
Time: 14.5s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) + \left(\sqrt{\sqrt[3]{x}} \cdot \left(\sqrt[3]{x} \cdot \sqrt{\sqrt[3]{x}}\right) + \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right)}\]

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 add-cube-cbrt30.0

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

  4. Applied cbrt-prod30.0

    \[\leadsto \color{blue}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}} - \sqrt[3]{x}\]
  5. Using strategy rm

  6. Applied flip3--30.0

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

  7. Taylor expanded around -inf 0.7

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

  9. Applied add-sqr-sqrt0.7

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

  10. Applied associate-*r*0.7

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

  11. Final simplification0.7

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

Reproduce

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

Details

Time bar (total: 13.2s)Debug log

sample66.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
8.0ms
(- (cbrt (+ x 1)) (cbrt x))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 25.8b

localize21.0ms

Local error

Found 3 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
8×add-sqr-sqrt
6×*-un-lft-identity
5×add-log-exp
4×add-cube-cbrt
4×cbrt-prod
3×add-exp-log
3×add-cbrt-cube
3×pow1
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 → 36
Calls
3 calls:
Slowest
7.0ms
(- (cbrt (+ x 1)) (cbrt x))
1.0ms
(cbrt (+ x 1))
0.0ms
(cbrt x)

series458.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
292.0ms
(cbrt x)
124.0ms
(- (cbrt (+ x 1)) (cbrt x))
41.0ms
(cbrt (+ x 1))

simplify931.0ms

Counts
20 → 45
Calls
20 calls:
Slowest
273.0ms
(- (+ (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)))))
209.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
190.0ms
(- (+ (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))))
70.0ms
(- (+ (* 1/3 (pow (/ 1 (pow x 2)) 1/3)) (pow (/ 1 x) -1/3)) (* 1/9 (pow (/ 1 (pow x 5)) 1/3)))
53.0ms
(- (+ (* 1/3 x) 1) (* 1/9 (pow x 2)))

prune532.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 11.4b

localize19.0ms

Local error

Found 4 expressions with local error:

8.6b
(- (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) (cbrt x))
0.6b
(cbrt x)
0.2b
(* (cbrt (+ x 1)) (cbrt (+ x 1)))
0.2b
(cbrt (+ x 1))

rewrite22.0ms

Algorithm
rewrite-expression-head
Rules
14×cbrt-div
9×add-cube-cbrt
9×*-un-lft-identity
9×pow1
9×add-sqr-sqrt
9×cbrt-prod
7×flip-+
7×flip3-+
6×add-log-exp
6×add-exp-log
6×associate-*l*
6×associate-*r*
6×pow1/3
4×add-cbrt-cube
4×frac-times
2×pow-prod-up
2×associate-*r/
2×associate-*l/
2×pow-prod-down
1×flip--
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×diff-log
1×flip3--
1×pow-plus
1×sub-neg
1×pow2
Counts
4 → 69
Calls
4 calls:
Slowest
15.0ms
(- (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) (cbrt x))
5.0ms
(* (cbrt (+ x 1)) (cbrt (+ x 1)))
1.0ms
(cbrt (+ x 1))
0.0ms
(cbrt x)

series453.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
273.0ms
(cbrt x)
92.0ms
(- (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) (cbrt x))
45.0ms
(* (cbrt (+ x 1)) (cbrt (+ x 1)))
43.0ms
(cbrt (+ x 1))

simplify1.5s

Counts
50 → 81
Calls
50 calls:
Slowest
251.0ms
(- (+ (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)))))
209.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
197.0ms
(- (+ (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))))
70.0ms
(- (+ (* 2/3 (pow (/ 1 x) 1/3)) (pow (/ -1 x) -2/3)) (* 1/9 (pow (/ 1 (pow x 4)) 1/3)))
69.0ms
(- (+ (* 2/3 x) 1) (* 1/9 (pow x 2)))

prune980.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 11.4b

localize37.0ms

Local error

Found 4 expressions with local error:

8.3b
(- (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3) (pow (cbrt x) 3))
0.7b
(pow (cbrt x) 3)
0.6b
(cbrt x)
0.6b
(cbrt x)

rewrite32.0ms

Algorithm
rewrite-expression-head
Rules
12×add-sqr-sqrt
8×*-un-lft-identity
6×add-log-exp
6×add-cube-cbrt
5×add-exp-log
5×pow1
4×cube-prod
4×add-cbrt-cube
4×unpow-prod-down
3×difference-of-squares
3×pow1/3
2×pow-pow
1×rem-cube-cbrt
1×distribute-lft-out--
1×flip--
1×pow-exp
1×unpow3
1×diff-log
1×flip3--
1×sub-neg
1×difference-cubes
1×pow-to-exp
1×cube-mult
Counts
4 → 53
Calls
4 calls:
Slowest
28.0ms
(- (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3) (pow (cbrt x) 3))
2.0ms
(pow (cbrt x) 3)
0.0ms
(cbrt x)
0.0ms
(cbrt x)

series875.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
291.0ms
(cbrt x)
281.0ms
(pow (cbrt x) 3)
226.0ms
(cbrt x)
76.0ms
(- (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3) (pow (cbrt x) 3))

simplify1.1s

Counts
30 → 65
Calls
30 calls:
Slowest
149.0ms
(/ (exp (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3)) (exp (pow (cbrt x) 3)))
135.0ms
(- (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3) (pow (cbrt x) 3))
132.0ms
(- (sqrt (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3)) (sqrt (pow (cbrt x) 3)))
129.0ms
(+ (sqrt (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3)) (pow (sqrt (cbrt x)) 3))
104.0ms
(- (sqrt (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3)) (pow (sqrt (cbrt x)) 3))

prune1.1s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.4b

localize25.0ms

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))

rewrite9.0ms

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×add-cbrt-cube
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 → 46
Calls
4 calls:
Slowest
7.0ms
(* (cbrt x) (cbrt x))
0.0ms
(cbrt x)
0.0ms
(cbrt x)
0.0ms
(cbrt x)

series1.2s

Counts
4 → 12
Calls
4 calls:
Slowest
347.0ms
(cbrt x)
314.0ms
(cbrt x)
288.0ms
(cbrt x)
284.0ms
(* (cbrt x) (cbrt x))

simplify59.0ms

Counts
25 → 58
Calls
25 calls:
Slowest
16.0ms
(* (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))
5.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
4.0ms
(pow (/ 1 x) -2/3)

prune758.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.3b

regimes74.0ms

Accuracy

0% (0.3b remaining)

Error of 0.7b against oracle of 0.4b and baseline of 0.7b

bsearch5.0ms

end0.0ms

sample2.8s

Algorithm
intervals