Average Error: 29.1 → 19.0
Time: 48.7s
Precision: 64
Internal Precision: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -9488381.159983333:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{elif}\;n \le -8.075163323171836 \cdot 10^{-307}:\\ \;\;\;\;\left(\log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}}\right) - \log \left(\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\right) + \left(\log \left(\sqrt[3]{\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}} \cdot \sqrt[3]{\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt[3]{\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \mathbf{elif}\;n \le 57004.62942959996:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \end{array}\]

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 n < -9488381.159983333 or 57004.62942959996 < n

    1. Initial program 44.2

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

    2. Taylor expanded around -inf 63.0

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

    3. Simplified31.9

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\frac{\frac{1}{x}}{n} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)}\]

    if -9488381.159983333 < n < -8.075163323171836e-307

    1. Initial program 0.7

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

    3. Applied add-log-exp0.8

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

    4. Applied add-log-exp0.8

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

    5. Applied diff-log0.8

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

    6. Simplified0.8

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

    8. Applied add-sqr-sqrt0.8

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

    9. Applied log-prod0.8

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

    11. Applied exp-diff0.8

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

    12. Applied sqrt-div0.8

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

    13. Applied log-div0.8

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

    15. Applied add-cube-cbrt0.8

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

    16. Applied log-prod0.8

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

    if -8.075163323171836e-307 < n < 57004.62942959996

    1. Initial program 23.9

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

    3. Applied add-exp-log24.3

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

    4. Applied pow-exp24.3

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

    5. Simplified1.5

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

  4. Final simplification19.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -9488381.159983333:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{elif}\;n \le -8.075163323171836 \cdot 10^{-307}:\\ \;\;\;\;\left(\log \left(\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}}\right) - \log \left(\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\right) + \left(\log \left(\sqrt[3]{\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}} \cdot \sqrt[3]{\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt[3]{\sqrt{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \mathbf{elif}\;n \le 57004.62942959996:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))

Details

Time bar (total: 47.3s)Debug log

sample334.0ms

Algorithm
intervals

simplify11.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

10.0ms
(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.0b

localize58.0ms

Local error

Found 3 expressions with local error:

1.5b
(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))
1.0b
(pow (+ x 1) (/ 1 n))
0.3b
(pow x (/ 1 n))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
3 → 74
Calls

3 calls. Slowest were:

19.0ms
(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))
2.0ms
(pow (+ x 1) (/ 1 n))
1.0ms
(pow x (/ 1 n))

series667.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

507.0ms
(- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))
94.0ms
(pow (+ x 1) (/ 1 n))
66.0ms
(pow x (/ 1 n))

simplify2.8s

Counts
69 → 83
Calls

69 calls. Slowest were:

562.0ms
(- (+ (/ (log -1) n) (+ 1 (/ 1 (* x n)))) (/ (log (/ -1 x)) n))
422.0ms
(- (+ (/ (log -1) (* x (pow n 2))) (/ 1 (* x n))) (+ (* 1/2 (/ 1 (* (pow x 2) n))) (/ (log (/ -1 x)) (* x (pow n 2)))))
335.0ms
(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))

prune854.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 16.7b

localize14.0ms

Local error

Found 4 expressions with local error:

1.5b
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
1.5b
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
1.0b
(pow (+ 1 x) (/ 1 n))
0.3b
(pow x (/ 1 n))

rewrite35.0ms

Algorithm
rewrite-expression-head
Counts
4 → 115
Calls

4 calls. Slowest were:

19.0ms
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
11.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
2.0ms
(pow (+ 1 x) (/ 1 n))

series577.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

201.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
185.0ms
(log (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))))
100.0ms
(pow (+ 1 x) (/ 1 n))
90.0ms
(pow x (/ 1 n))

simplify5.1s

Counts
103 → 127
Calls

103 calls. Slowest were:

572.0ms
(- (+ (/ (log -1) n) (+ 1 (/ 1 (* x n)))) (/ (log (/ -1 x)) n))
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)))))
345.0ms
(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))

prune1.4s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 16.7b

localize17.0ms

Local error

Found 4 expressions with local error:

1.5b
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
1.5b
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
1.5b
(log (sqrt (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n))))))
1.5b
(log (sqrt (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n))))))

rewrite63.0ms

Algorithm
rewrite-expression-head
Counts
4 → 160
Calls

4 calls. Slowest were:

22.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
20.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
9.0ms
(log (sqrt (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n))))))

series1.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

294.0ms
(log (sqrt (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n))))))
292.0ms
(log (sqrt (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n))))))
243.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
186.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))

simplify7.0s

Counts
162 → 172
Calls

162 calls. Slowest were:

489.0ms
(- (+ (* 1/2 (/ (log -1) (* x (pow n 2)))) (* 1/2 (/ 1 (* x n)))) (+ (* 1/4 (/ 1 (* (pow x 2) n))) (* 1/2 (/ (log (/ -1 x)) (* x (pow n 2))))))
458.0ms
(- (+ (* 1/2 (/ (log -1) (* x (pow n 2)))) (* 1/2 (/ 1 (* x n)))) (+ (* 1/4 (/ 1 (* (pow x 2) n))) (* 1/2 (/ (log (/ -1 x)) (* x (pow n 2))))))
416.0ms
(- (/ 1 (* x n)) (+ (/ (log (/ 1 x)) (* x (pow n 2))) (* 1/2 (/ 1 (* (pow x 2) n)))))

prune3.2s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 16.7b

localize46.0ms

Local error

Found 4 expressions with local error:

1.5b
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
1.5b
(- (log (sqrt (exp (pow (+ 1 x) (/ 1 n))))) (log (sqrt (exp (pow x (/ 1 n))))))
1.5b
(log (sqrt (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n))))))
1.0b
(pow (+ 1 x) (/ 1 n))

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 161
Calls

4 calls. Slowest were:

15.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
13.0ms
(log (sqrt (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n))))))
9.0ms
(- (log (sqrt (exp (pow (+ 1 x) (/ 1 n))))) (log (sqrt (exp (pow x (/ 1 n))))))

series1.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

350.0ms
(- (log (sqrt (exp (pow (+ 1 x) (/ 1 n))))) (log (sqrt (exp (pow x (/ 1 n))))))
332.0ms
(log (sqrt (exp (- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n))))))
257.0ms
(- (pow (+ 1 x) (/ 1 n)) (pow x (/ 1 n)))
69.0ms
(pow (+ 1 x) (/ 1 n))

simplify6.9s

Counts
168 → 173
Calls

168 calls. Slowest were:

615.0ms
(- (+ (* 1/2 (/ (log -1) (* x (pow n 2)))) (* 1/2 (/ 1 (* x n)))) (+ (* 1/4 (/ 1 (* (pow x 2) n))) (* 1/2 (/ (log (/ -1 x)) (* x (pow n 2))))))
557.0ms
(- (+ (* 1/2 (/ (log -1) (* x (pow n 2)))) (* 1/2 (/ 1 (* x n)))) (+ (* 1/4 (/ 1 (* (pow x 2) n))) (* 1/2 (/ (log (/ -1 x)) (* x (pow n 2))))))
511.0ms
(- (+ (/ (log -1) n) (+ 1 (/ 1 (* x n)))) (/ (log (/ -1 x)) n))

prune3.7s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 16.5b

regimes418.0ms

Accuracy

87.1% (1.5b remaining)

Error of 19.0b against oracle of 17.5b and baseline of 29.3b

bsearch782.0ms

end0.0ms

sample11.3s

Algorithm
intervals