Average Error: 30.0 → 22.4
Time: 45.1s
Precision: 64
Internal Precision: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.2293321380368027 \cdot 10^{-12}:\\ \;\;\;\;\sqrt[3]{\left(\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \log \left(e^{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.3767336758462545 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \frac{1}{x \cdot n}\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \left(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 n) < -1.2293321380368027e-12

    1. Initial program 1.6

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    2. Using strategy rm

    3. Applied add-log-exp1.6

      \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    4. Using strategy rm

    5. Applied add-log-exp1.8

      \[\leadsto \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]

    6. Applied diff-log1.8

      \[\leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]

    7. Simplified1.8

      \[\leadsto \log \color{blue}{\left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube1.8

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt1.8

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

    12. Applied add-sqr-sqrt1.8

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

    13. Applied difference-of-squares1.8

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

    if -1.2293321380368027e-12 < (/ 1 n) < 3.3767336758462545e-08

    1. Initial program 45.1

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    2. Using strategy rm

    3. Applied add-log-exp45.1

      \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    4. Using strategy rm

    5. Applied add-log-exp45.1

      \[\leadsto \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]

    6. Applied diff-log45.1

      \[\leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]

    7. Simplified45.1

      \[\leadsto \log \color{blue}{\left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]

    8. Taylor expanded around inf 31.9

      \[\leadsto \color{blue}{\frac{1}{x \cdot n} - \left(\frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot n}\right)}\]

    9. Simplified31.9

      \[\leadsto \color{blue}{\frac{\log x}{n \cdot \left(n \cdot x\right)} + \left(\frac{\frac{\frac{-1}{2}}{x}}{n \cdot x} + \frac{1}{n \cdot x}\right)}\]

    if 3.3767336758462545e-08 < (/ 1 n)

    1. Initial program 25.6

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    2. Using strategy rm

    3. Applied add-log-exp25.6

      \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    4. Using strategy rm

    5. Applied add-log-exp25.6

      \[\leadsto \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]

    6. Applied diff-log25.6

      \[\leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]

    7. Simplified25.6

      \[\leadsto \log \color{blue}{\left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube25.6

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt25.6

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

    12. Applied add-sqr-sqrt25.6

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

    13. Applied difference-of-squares25.6

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

    14. Applied exp-prod25.6

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

    15. Applied log-pow25.6

      \[\leadsto \sqrt[3]{\left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \color{blue}{\left(\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.2293321380368027 \cdot 10^{-12}:\\ \;\;\;\;\sqrt[3]{\left(\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \log \left(e^{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 3.3767336758462545 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \frac{1}{x \cdot n}\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \left(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 1137770271 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))

Details

Time bar (total: 42.2s)Debug log

sample550.0ms

Algorithm
intervals
Results
95×155.0ms(body real 1280)
35×75.0ms(body exit 10240)
46×53.0ms(body real 640)
29×46.0ms(body nan 1280)
41×34.0ms(body real 320)
152×34.0ms(body real 80)
27×33.0ms(body nan 640)
159×27.0ms(body nan 80)
618×18.0ms(pre true 80)
14×11.0ms(body nan 320)
15×8.0ms(body real 160)
5×2.0ms(body nan 160)

simplify38.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
38.0ms
(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 29.6b

localize46.0ms

Local error

Found 3 expressions with local error:

3.0b
(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))
1.2b
(pow (+ x 1) (/ 1 n))
0.7b
(pow x (/ 1 n))

rewrite27.0ms

Algorithm
rewrite-expression-head
Rules
10×add-sqr-sqrt
8×*-un-lft-identity
8×pow-unpow
6×add-cube-cbrt
5×add-log-exp
4×add-exp-log
4×unpow-prod-down
4×pow1
3×add-cbrt-cube
2×difference-of-squares
2×div-inv
2×pow-to-exp
1×distribute-lft-out--
1×flip--
1×pow-exp
1×diff-log
1×flip3--
1×pow-pow
1×sub-neg
Counts
3 → 44
Calls
3 calls:
Slowest
20.0ms
(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))
4.0ms
(pow (+ x 1) (/ 1 n))
2.0ms
(pow x (/ 1 n))

series568.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
334.0ms
(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))
124.0ms
(pow (+ x 1) (/ 1 n))
110.0ms
(pow x (/ 1 n))

simplify2.5s

Counts
28 → 53
Calls
28 calls:
Slowest
1.1s
(- (+ (/ (log -1) n) (+ 1 (/ 1 (* x n)))) (/ (log (/ -1 x)) n))
412.0ms
(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
363.0ms
(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))
339.0ms
(- (+ (* 1/2 (/ (pow (log (/ 1 x)) 2) (pow n 2))) 1) (/ (log (/ 1 x)) n))
140.0ms
(- (+ (/ (log -1) n) (+ (* 1/2 (/ (pow (log -1) 2) (pow n 2))) (+ (* 1/2 (/ (pow (log (/ -1 x)) 2) (pow n 2))) 1))) (+ (/ (log (/ -1 x)) n) (/ (* (log (/ -1 x)) (log -1)) (pow n 2))))

prune954.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 20.8b

localize27.0ms

Local error

Found 4 expressions with local error:

3.5b
(- (log (exp (pow (+ x 1) (/ 1 n)))) (pow x (/ 1 n)))
1.2b
(pow (+ x 1) (/ 1 n))
0.7b
(pow x (/ 1 n))
0.0b
(exp (pow (+ x 1) (/ 1 n)))

rewrite79.0ms

Algorithm
rewrite-expression-head
Rules
13×*-un-lft-identity
12×add-sqr-sqrt
10×add-cube-cbrt
8×pow-unpow
6×add-log-exp
6×unpow-prod-down
6×exp-prod
6×pow1
5×add-exp-log
4×add-cbrt-cube
3×associate--l+
3×log-prod
2×distribute-lft-out--
2×div-inv
2×pow-to-exp
1×rem-exp-log
1×difference-of-squares
1×flip--
1×log-pow
1×pow-exp
1×diff-log
1×flip3--
1×pow-pow
1×sub-neg
Counts
4 → 61
Calls
4 calls:
Slowest
61.0ms
(- (log (exp (pow (+ x 1) (/ 1 n)))) (pow x (/ 1 n)))
10.0ms
(exp (pow (+ x 1) (/ 1 n)))
4.0ms
(pow (+ x 1) (/ 1 n))
2.0ms
(pow x (/ 1 n))

series859.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
385.0ms
(- (log (exp (pow (+ x 1) (/ 1 n)))) (pow x (/ 1 n)))
241.0ms
(exp (pow (+ x 1) (/ 1 n)))
122.0ms
(pow (+ x 1) (/ 1 n))
110.0ms
(pow x (/ 1 n))

simplify3.0s

Counts
40 → 73
Calls
40 calls:
Slowest
870.0ms
(- (+ (/ (log -1) n) (+ 1 (/ 1 (* x n)))) (/ (log (/ -1 x)) n))
508.0ms
(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
392.0ms
(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))
325.0ms
(- (+ (* 1/2 (/ (pow (log (/ 1 x)) 2) (pow n 2))) 1) (/ (log (/ 1 x)) n))
296.0ms
(- (+ E (+ (/ (* E (log -1)) n) (/ E (* x n)))) (/ (* (log (/ -1 x)) E) n))

prune1.6s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 20.7b

localize43.0ms

Local error

Found 4 expressions with local error:

3.0b
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
3.0b
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
1.2b
(pow (+ 1 x) (/ 1 n))
0.7b
(pow x (/ 1 n))

rewrite79.0ms

Algorithm
rewrite-expression-head
Rules
17×add-sqr-sqrt
13×*-un-lft-identity
9×add-cube-cbrt
8×pow-unpow
7×log-pow
6×add-log-exp
6×exp-prod
6×pow1
5×add-exp-log
5×unpow-prod-down
5×log-prod
4×difference-of-squares
4×add-cbrt-cube
2×distribute-lft-out--
2×div-inv
2×exp-sum
2×sub-neg
2×pow-to-exp
1×flip--
1×pow-exp
1×rem-log-exp
1×diff-log
1×exp-diff
1×flip3--
1×pow-pow
1×log-div
Counts
4 → 65
Calls
4 calls:
Slowest
50.0ms
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
21.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
4.0ms
(pow (+ 1 x) (/ 1 n))
2.0ms
(pow x (/ 1 n))

series985.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
368.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
350.0ms
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
153.0ms
(pow (+ 1 x) (/ 1 n))
113.0ms
(pow x (/ 1 n))

simplify3.3s

Counts
44 → 77
Calls
44 calls:
Slowest
847.0ms
(- (+ (/ (log -1) n) (+ 1 (/ 1 (* x n)))) (/ (log (/ -1 x)) n))
415.0ms
(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
414.0ms
(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
378.0ms
(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))
365.0ms
(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))

prune1.7s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 20.7b

localize26.0ms

Local error

Found 4 expressions with local error:

3.0b
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
3.0b
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
3.0b
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
3.0b
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))

rewrite149.0ms

Algorithm
rewrite-expression-head
Rules
26×add-sqr-sqrt
21×log-pow
18×*-un-lft-identity
18×exp-prod
15×log-prod
10×add-cube-cbrt
8×difference-of-squares
7×pow1
6×add-log-exp
6×exp-sum
4×distribute-lft-out--
4×add-exp-log
4×add-cbrt-cube
4×unpow-prod-down
4×sub-neg
3×rem-log-exp
3×exp-diff
3×log-div
1×flip--
1×diff-log
1×flip3--
Counts
4 → 78
Calls
4 calls:
Slowest
43.0ms
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
41.0ms
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
41.0ms
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
20.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))

series1.4s

Counts
4 → 12
Calls
4 calls:
Slowest
374.0ms
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
358.0ms
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
348.0ms
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
347.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))

simplify3.9s

Counts
57 → 90
Calls
57 calls:
Slowest
469.0ms
(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
449.0ms
(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
417.0ms
(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
409.0ms
(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
388.0ms
(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))

prune2.5s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 20.7b

regimes655.0ms

Accuracy

84.7% (1.4b remaining)

Error of 22.4b against oracle of 21.0b and baseline of 30.0b

bsearch14.0ms

end0.0ms

sample17.1s

Algorithm
intervals
Results
3155×5.2s(body real 1280)
1642×2.5s(body real 640)
1146×2.4s(body exit 10240)
4710×1.3s(body real 80)
663×1.2s(body nan 1280)
5131×972.0ms(body nan 80)
634×813.0ms(body nan 640)
909×789.0ms(body real 320)
18954×735.0ms(pre true 80)
358×296.0ms(body nan 320)
428×225.0ms(body real 160)
178×88.0ms(body nan 160)