Average Error: 30.1 → 0.7
Time: 21.8s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) + \left(\left(\sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.1

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

  3. Applied add-cube-cbrt30.1

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

  4. Applied cbrt-prod30.1

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

  6. Applied flip3--30.1

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

  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

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 1042149663 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

Time bar (total: 19.3s)Debug log

sample139.0ms

Algorithm
intervals
Results
472×(pre true 80)
215×(body nan 80)
141×(body real 80)
52×(body real 1280)
32×(body real 640)
27×(body real 320)
5×(body real 160)

simplify10.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
9.0ms
(- (cbrt (+ x 1)) (cbrt x))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 27.6b

localize30.0ms

Local error

Found 3 expressions with local error:

2.2b
(- (cbrt (+ x 1)) (cbrt x))
0.5b
(cbrt x)
0.3b
(cbrt (+ x 1))

rewrite15.0ms

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
12.0ms
(- (cbrt (+ x 1)) (cbrt x))
1.0ms
(cbrt (+ x 1))
0.0ms
(cbrt x)

series692.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
453.0ms
(cbrt x)
174.0ms
(- (cbrt (+ x 1)) (cbrt x))
64.0ms
(cbrt (+ x 1))

simplify1.4s

Counts
20 → 45
Calls
20 calls:
Slowest
383.0ms
(- (+ (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)))))
344.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
278.0ms
(- (+ (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))))
102.0ms
(- (+ (* 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)))
99.0ms
(- (+ (* 1/3 x) 1) (* 1/9 (pow x 2)))

prune684.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 13.4b

localize37.0ms

Local error

Found 4 expressions with local error:

10.9b
(- (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) (cbrt x))
0.5b
(cbrt x)
0.3b
(cbrt (+ x 1))
0.3b
(cbrt (+ x 1))

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
6×add-log-exp
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
6×cbrt-prod
4×add-exp-log
4×add-cbrt-cube
4×cbrt-div
4×pow1
3×pow1/3
2×flip-+
2×flip3-+
1×flip--
1×diff-log
1×flip3--
1×sub-neg
Counts
4 → 46
Calls
4 calls:
Slowest
34.0ms
(- (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) (cbrt x))
1.0ms
(cbrt (+ x 1))
1.0ms
(cbrt (+ x 1))
0.0ms
(cbrt x)

series740.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
444.0ms
(cbrt x)
163.0ms
(- (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) (cbrt x))
68.0ms
(cbrt (+ x 1))
64.0ms
(cbrt (+ x 1))

simplify1.9s

Counts
23 → 58
Calls
23 calls:
Slowest
393.0ms
(- (+ (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)))))
344.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
285.0ms
(- (+ (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))))
281.0ms
(- (+ (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))))
100.0ms
(- (+ (* 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)))

prune1.1s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 13.4b

localize89.0ms

Local error

Found 4 expressions with local error:

10.3b
(- (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3) (pow (cbrt x) 3))
0.8b
(pow (cbrt x) 3)
0.5b
(cbrt x)
0.5b
(cbrt x)

rewrite40.0ms

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
36.0ms
(- (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)

series1.5s

Counts
4 → 12
Calls
4 calls:
Slowest
477.0ms
(pow (cbrt x) 3)
434.0ms
(cbrt x)
433.0ms
(cbrt x)
119.0ms
(- (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3) (pow (cbrt x) 3))

simplify1.6s

Counts
30 → 65
Calls
30 calls:
Slowest
202.0ms
(/ (exp (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3)) (exp (pow (cbrt x) 3)))
173.0ms
(- (sqrt (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3)) (pow (sqrt (cbrt x)) 3))
173.0ms
(- (sqrt (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3)) (pow (sqrt (cbrt x)) 3))
158.0ms
(- (sqrt (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3)) (sqrt (pow (cbrt x) 3)))
157.0ms
(- (pow (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (cbrt (+ x 1)))) 3) (pow (cbrt x) 3))

prune1.5s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.4b

localize26.0ms

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))

rewrite11.0ms

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
8.0ms
(* (cbrt x) (cbrt x))
0.0ms
(cbrt x)
0.0ms
(cbrt x)
0.0ms
(cbrt x)

series1.8s

Counts
4 → 12
Calls
4 calls:
Slowest
469.0ms
(* (cbrt x) (cbrt x))
434.0ms
(cbrt x)
434.0ms
(cbrt x)
430.0ms
(cbrt x)

simplify105.0ms

Counts
25 → 58
Calls
25 calls:
Slowest
21.0ms
(* (pow (pow x 2) 1/3) (pow (cbrt -1) 2))
11.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
11.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
11.0ms
(* (pow (* x -1) 1/3) (cbrt -1))
6.0ms
(pow (/ 1 x) -1/3)

prune1.1s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.3b

regimes306.0ms

Accuracy

0% (0.3b remaining)

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

bsearch6.0ms

end0.0ms

sample4.3s

Algorithm
intervals
Results
16095×(pre true 80)
8094×(body nan 80)
4038×(body real 80)
1722×(body real 1280)
1293×(body real 640)
638×(body real 320)
310×(body real 160)