Derivation

Split input into 2 regimes
if n < -964894.1011799229 or 7.586550074092197 < n
1. Initial program 44.5
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-log-exp44.5
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
4. Applied add-log-exp44.5
  \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
5. Applied diff-log44.5
  \[\leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
6. Simplified44.5
  \[\leadsto \log \color{blue}{\left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
7. Taylor expanded around inf 32.7
  \[\leadsto \color{blue}{\frac{1}{x \cdot n} - \left(\frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot n}\right)}\]
8. Simplified32.7
  \[\leadsto \color{blue}{\frac{\log x}{n \cdot \left(n \cdot x\right)} + \left(\frac{\frac{\frac{-1}{2}}{x}}{n \cdot x} + \frac{1}{n \cdot x}\right)}\]
if -964894.1011799229 < n < 7.586550074092197
1. Initial program 7.7
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-log-exp7.9
  \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
4. Applied add-log-exp7.8
  \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
5. Applied diff-log7.8
  \[\leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
6. Simplified7.8
  \[\leadsto \log \color{blue}{\left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
7. Using strategy rm
8. Applied add-cube-cbrt7.8
  \[\leadsto \log \left(e^{\color{blue}{\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\]
9. Applied exp-prod7.8
  \[\leadsto \log \color{blue}{\left({\left(e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^{\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right)}\]
10. Applied log-pow7.8
  \[\leadsto \color{blue}{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}\]
11. Using strategy rm
12. Applied add-log-exp7.8
  \[\leadsto \sqrt[3]{\color{blue}{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\]
13. Using strategy rm
14. Applied add-sqr-sqrt7.8
  \[\leadsto \sqrt[3]{\log \color{blue}{\left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\]
15. Applied log-prod7.8
  \[\leadsto \sqrt[3]{\color{blue}{\log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\]
Recombined 2 regimes into one program.
Final simplification22.2
\[\leadsto \begin{array}{l} \mathbf{if}\;n \le -964894.1011799229 \lor \neg \left(n \le 7.586550074092197\right):\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]

Details

sample329.0ms

Algorithm

intervals

Results

96.0ms	93×	body	1280	valid
50.0ms	28×	body	640	nan
46.0ms	54×	body	640	valid
34.0ms	41×	body	10240	exit
22.0ms	146×	body	80	valid
16.0ms	14×	body	1280	nan
16.0ms	134×	body	80	nan
15.0ms	26×	body	320	valid
7.0ms	14×	body	160	valid
7.0ms	13×	body	320	nan
2.0ms	5×	body	160	nan

simplify6.0ms

Counts

1 → 1

Calls

1 calls:

Slowest

6.0ms

(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 32.4b

localize31.0ms

Local error

Found 3 expressions with local error:

2.4b	(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))
1.0b	(pow x (/ 1 n))
0.8b	(pow (+ x 1) (/ 1 n))

rewrite19.0ms

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

14.0ms	(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))
3.0ms	(pow (+ x 1) (/ 1 n))
1.0ms	(pow x (/ 1 n))

series348.0ms

Counts

3 → 9

Calls

3 calls:

Slowest

215.0ms	(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))
81.0ms	(pow (+ x 1) (/ 1 n))
51.0ms	(pow x (/ 1 n))

simplify1.4s

Counts

28 → 53

Calls

28 calls:

Slowest

504.0ms	(- (+ (/ (log -1) n) (+ 1 (/ 1 (* x n)))) (/ (log (/ -1 x)) n))
260.0ms	(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
246.0ms	(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))
199.0ms	(- (+ (* 1/2 (/ (pow (log (/ 1 x)) 2) (pow n 2))) 1) (/ (log (/ 1 x)) n))
61.0ms	(- (+ (/ (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))))

prune528.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 22.2b

localize15.0ms

Local error

Found 4 expressions with local error:

2.4b	(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
2.4b	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
1.0b	(pow x (/ 1 n))
0.8b	(pow (+ 1 x) (/ 1 n))

rewrite28.0ms

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

17.0ms	(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
8.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
1.0ms	(pow (+ 1 x) (/ 1 n))
1.0ms	(pow x (/ 1 n))

series531.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

204.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
193.0ms	(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
71.0ms	(pow (+ 1 x) (/ 1 n))
63.0ms	(pow x (/ 1 n))

simplify2.3s

Counts

44 → 77

Calls

44 calls:

Slowest

552.0ms	(- (+ (/ (log -1) n) (+ 1 (/ 1 (* x n)))) (/ (log (/ -1 x)) n))
308.0ms	(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
304.0ms	(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
271.0ms	(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))
224.0ms	(- (+ (* 1/2 (/ (pow (log (/ 1 x)) 2) (pow n 2))) 1) (/ (log (/ 1 x)) n))

prune885.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 22.2b

localize17.0ms

Local error

Found 4 expressions with local error:

2.4b	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
2.4b	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
2.4b	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
1.9b	(log (exp (* (cbrt (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))) (cbrt (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))))

rewrite35.0ms

Algorithm

rewrite-expression-head

Rules

17×	`add-sqr-sqrt`
11×	`add-log-exp`
11×	`*-un-lft-identity`
6×	`difference-of-squares`
5×	`add-cube-cbrt`
5×	`pow1`
4×	`add-exp-log`
4×	`add-cbrt-cube`
3×	`distribute-lft-out--`
3×	`flip--`
3×	`log-pow`
3×	`diff-log`
3×	`flip3--`
3×	`unpow-prod-down`
3×	`sub-neg`
3×	`log-prod`
1×	`rem-log-exp`
1×	`exp-to-pow`
1×	`exp-prod`

Counts

4 → 59

Calls

4 calls:

Slowest

14.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
10.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
8.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
2.0ms	(log (exp (* (cbrt (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))) (cbrt (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))))

series1.1s

Counts

4 → 12

Calls

4 calls:

Slowest

495.0ms	(log (exp (* (cbrt (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))) (cbrt (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))))
208.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
189.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
174.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))

simplify2.3s

Counts

36 → 71

Calls

36 calls:

Slowest

274.0ms	(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
266.0ms	(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))
263.0ms	(- (+ (* 2/3 (/ (* (exp (* 1/3 (+ (* 2 (log (/ -1 x))) (* 2 (log (/ -1 n)))))) (log -1)) n)) (exp (* 1/3 (+ (* 2 (log (/ -1 x))) (* 2 (log (/ -1 n))))))) (+ (* 2/3 (/ (* (log (/ -1 x)) (exp (* 1/3 (+ (* 2 (log (/ -1 x))) (* 2 (log (/ -1 n))))))) n)) (* 1/3 (/ (exp (* 1/3 (+ (* 2 (log (/ -1 x))) (* 2 (log (/ -1 n)))))) x))))
260.0ms	(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
246.0ms	(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))

prune936.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 22.2b

localize14.0ms

Local error

Found 4 expressions with local error:

2.4b	(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
2.4b	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
2.4b	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
2.4b	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))

rewrite58.0ms

Algorithm

rewrite-expression-head

Rules

22×	`add-sqr-sqrt`
14×	`*-un-lft-identity`
10×	`add-log-exp`
8×	`difference-of-squares`
7×	`log-pow`
6×	`add-cube-cbrt`
6×	`exp-prod`
5×	`log-prod`
5×	`pow1`
4×	`distribute-lft-out--`
4×	`add-exp-log`
4×	`add-cbrt-cube`
4×	`unpow-prod-down`
4×	`sub-neg`
3×	`flip--`
3×	`diff-log`
3×	`flip3--`
2×	`exp-sum`
1×	`rem-log-exp`
1×	`exp-diff`
1×	`log-div`

Counts

4 → 66

Calls

4 calls:

Slowest

23.0ms	(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
16.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
9.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
8.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))

series846.0ms

Counts

4 → 12

Calls

4 calls:

Slowest

233.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
217.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
213.0ms	(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
182.0ms	(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))

simplify2.4s

Counts

43 → 78

Calls

43 calls:

Slowest

313.0ms	(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
285.0ms	(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
285.0ms	(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))
250.0ms	(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
235.0ms	(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))

prune1.2s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 22.2b

regimes254.0ms

Accuracy

82.9% (1.4b remaining)

Error of 22.2b against oracle of 20.8b and baseline of 29.0b

bsearch429.0ms

end0.0ms

sample10.6s

Algorithm

intervals

Results

3.6s	3133×	body	1280	valid
1.3s	1134×	body	10240	exit
1.3s	1543×	body	640	valid
1.2s	4787×	body	80	valid
742.0ms	640×	body	1280	nan
599.0ms	5030×	body	80	nan
537.0ms	661×	body	640	nan
511.0ms	831×	body	320	valid
253.0ms	377×	body	320	nan
153.0ms	473×	body	160	valid
102.0ms	166×	body	160	nan

Error

Try it out

Derivation

`if n < -964894.1011799229 or 7.586550074092197 < n`

`if -964894.1011799229 < n < 7.586550074092197`

Reproduce

Details

sample329.0ms

simplify6.0ms

prune6.0ms

localize31.0ms

rewrite19.0ms

series348.0ms

simplify1.4s

prune528.0ms

localize15.0ms

rewrite28.0ms

series531.0ms

simplify2.3s

prune885.0ms

localize17.0ms

rewrite35.0ms

series1.1s

simplify2.3s

prune936.0ms

localize14.0ms

rewrite58.0ms

series846.0ms

simplify2.4s

prune1.2s

regimes254.0ms

bsearch429.0ms

end0.0ms

sample10.6s

In
Out