Average Error: 29.2 → 2.2
Time: 27.3s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 6894.936877853371:\\ \;\;\;\;e^{\log \left(\frac{{\left(\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right)\right) \cdot \sqrt[3]{1 + x}}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right)\right) \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right)\right) \cdot \sqrt[3]{1 + x}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right)\right) \cdot \sqrt[3]{1 + x}} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right)}\\ \mathbf{elif}\;x \le 1.3607407033606098 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right) + \sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\log \frac{1}{3} - \log x\right) + \frac{\frac{7}{54}}{x \cdot x}\right) + \left(\frac{1}{3} \cdot \log x - \frac{\frac{1}{3}}{x}\right)}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < 6894.936877853371

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.2

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

    5. Applied add-cube-cbrt0.2

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

    7. Applied add-exp-log0.2

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

    9. Applied flip3--0.2

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

    if 6894.936877853371 < x < 1.3607407033606098e+154

    1. Initial program 59.5

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

    3. Applied add-cbrt-cube59.2

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

    5. Applied add-cube-cbrt58.9

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

    7. Applied add-exp-log58.8

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

    8. Taylor expanded around inf 5.1

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + \frac{5}{81} \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    9. Simplified0.9

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \sqrt[3]{\frac{1}{x \cdot x}} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right) + \sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}}\]

    if 1.3607407033606098e+154 < x

    1. Initial program 61.0

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

    3. Applied add-cbrt-cube61.2

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

    5. Applied add-cube-cbrt61.4

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

    7. Applied add-exp-log61.4

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

    8. Taylor expanded around inf 6.9

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{3} \cdot {x}^{\frac{1}{3}}\right) + \left(\log \left(\frac{1}{x}\right) + \frac{7}{54} \cdot \frac{1}{{x}^{2}}\right)\right) - \frac{1}{3} \cdot \frac{1}{x}}}\]

    9. Simplified7.6

      \[\leadsto e^{\color{blue}{\left(\log x \cdot \frac{1}{3} - \frac{\frac{1}{3}}{x}\right) + \left(\left(\log \frac{1}{3} - \log x\right) + \frac{\frac{7}{54}}{x \cdot x}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 6894.936877853371:\\ \;\;\;\;e^{\log \left(\frac{{\left(\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right)\right) \cdot \sqrt[3]{1 + x}}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right)\right) \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right)\right) \cdot \sqrt[3]{1 + x}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right)\right) \cdot \sqrt[3]{1 + x}} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right)}\\ \mathbf{elif}\;x \le 1.3607407033606098 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right) + \sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\log \frac{1}{3} - \log x\right) + \frac{\frac{7}{54}}{x \cdot x}\right) + \left(\frac{1}{3} \cdot \log x - \frac{\frac{1}{3}}{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019018 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

Time bar (total: 26.2s)Debug log

sample51.0ms

Algorithm
intervals
Results
11.0ms284×body80nan
9.0ms44×body1280valid
7.0ms47×body640valid
5.0ms126×body80valid
3.0ms27×body320valid
1.0ms13×body160valid

simplify4.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
4.0ms
(- (cbrt (+ x 1)) (cbrt x))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 31.0b

localize10.0ms

Local error

Found 3 expressions with local error:

3.4b
(- (cbrt (+ x 1)) (cbrt x))
0.6b
(cbrt x)
0.2b
(cbrt (+ x 1))

rewrite7.0ms

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
6.0ms
(- (cbrt (+ x 1)) (cbrt x))
1.0ms
(cbrt (+ x 1))
0.0ms
(cbrt x)

series429.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
292.0ms
(cbrt x)
92.0ms
(- (cbrt (+ x 1)) (cbrt x))
45.0ms
(cbrt (+ x 1))

simplify952.0ms

Counts
20 → 45
Calls
20 calls:
Slowest
290.0ms
(- (+ (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)))))
244.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
144.0ms
(- (+ (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))))
77.0ms
(- (+ (* 1/3 x) 1) (* 1/9 (pow x 2)))
55.0ms
(- (+ (* 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)))

prune491.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 14.1b

localize19.0ms

Local error

Found 4 expressions with local error:

3.4b
(- (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1)))) (cbrt x))
0.6b
(cbrt x)
0.3b
(* (cbrt (+ x 1)) (cbrt (+ x 1)))
0.2b
(cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1))))

rewrite48.0ms

Algorithm
rewrite-expression-head
Rules
92×cbrt-div
33×flip-+
33×flip3-+
32×frac-times
16×associate-*l/
10×associate-*r/
10×*-un-lft-identity
10×add-sqr-sqrt
9×pow1
8×add-cube-cbrt
7×cbrt-prod
6×add-log-exp
6×add-exp-log
6×associate-*l*
6×associate-*r*
6×pow1/3
4×add-cbrt-cube
2×pow-prod-up
2×pow-prod-down
1×difference-of-squares
1×distribute-lft-out--
1×flip--
1×cbrt-unprod
1×rem-cbrt-cube
1×*-commutative
1×prod-exp
1×pow3
1×diff-log
1×flip3--
1×pow-plus
1×sub-neg
1×pow2
Counts
4 → 94
Calls
4 calls:
Slowest
24.0ms
(- (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1)))) (cbrt x))
12.0ms
(cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1))))
10.0ms
(* (cbrt (+ x 1)) (cbrt (+ x 1)))
0.0ms
(cbrt x)

series452.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
281.0ms
(cbrt x)
82.0ms
(- (cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1)))) (cbrt x))
49.0ms
(* (cbrt (+ x 1)) (cbrt (+ x 1)))
39.0ms
(cbrt (* (* (cbrt (+ x 1)) (cbrt (+ x 1))) (cbrt (+ x 1))))

simplify2.5s

Counts
76 → 106
Calls
76 calls:
Slowest
206.0ms
(- (+ (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)))))
203.0ms
(- (+ (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))))
182.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
69.0ms
(cbrt (* (* (cbrt (+ (pow x 3) (pow 1 3))) (cbrt (- (* x x) (* 1 1)))) (cbrt (+ x 1))))
68.0ms
(cbrt (* (* (cbrt (- (* x x) (* 1 1))) (cbrt (+ (pow x 3) (pow 1 3)))) (cbrt (+ x 1))))

prune998.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 14.1b

localize18.0ms

Local error

Found 4 expressions with local error:

9.3b
(- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x))
0.6b
(cbrt x)
0.3b
(* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))))
0.3b
(* (cbrt (+ x 1)) (cbrt (+ x 1)))

rewrite284.0ms

Algorithm
rewrite-expression-head
Rules
372×cbrt-div
140×frac-times
114×flip-+
114×flip3-+
46×associate-*r/
46×associate-*l/
14×pow1
12×*-un-lft-identity
12×add-sqr-sqrt
10×add-cube-cbrt
10×associate-*l*
10×associate-*r*
9×pow1/3
8×add-exp-log
8×cbrt-prod
6×add-log-exp
4×pow-prod-up
4×pow-prod-down
4×add-cbrt-cube
2×cbrt-unprod
2×*-commutative
2×prod-exp
2×pow-plus
2×pow2
1×difference-of-squares
1×distribute-lft-out--
1×flip--
1×diff-log
1×flip3--
1×sub-neg
Counts
4 → 162
Calls
4 calls:
Slowest
245.0ms
(- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x))
24.0ms
(* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))))
5.0ms
(* (cbrt (+ x 1)) (cbrt (+ x 1)))
0.0ms
(cbrt x)

series452.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
242.0ms
(cbrt x)
126.0ms
(- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x))
49.0ms
(* (cbrt (+ x 1)) (cbrt (+ x 1)))
33.0ms
(* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))))

simplify7.9s

Counts
208 → 174
Calls
208 calls:
Slowest
300.0ms
(- (+ (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)))))
217.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
81.0ms
(- (+ (* 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)))
81.0ms
(* (cbrt (* (cbrt (+ (pow x 3) (pow 1 3))) (cbrt (- (* x x) (* 1 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ (pow x 3) (pow 1 3))))))
79.0ms
(* (cbrt (* (cbrt (+ (pow x 3) (pow 1 3))) (cbrt (+ (pow x 3) (pow 1 3))))) (cbrt (* (cbrt (- (* x x) (* 1 1))) (cbrt (+ (pow x 3) (pow 1 3))))))

prune3.1s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 14.1b

localize10.0ms

Local error

Found 4 expressions with local error:

9.3b
(- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x))
8.2b
(log (- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x)))
0.6b
(cbrt x)
0.4b
(exp (log (- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x))))

rewrite1.4s

Algorithm
rewrite-expression-head
Rules
13×*-un-lft-identity
13×add-sqr-sqrt
10×log-prod
7×add-cube-cbrt
6×add-log-exp
6×pow1
5×add-exp-log
5×exp-sum
4×add-cbrt-cube
4×exp-prod
4×log-div
3×difference-of-squares
3×distribute-lft-out--
3×flip--
3×flip3--
2×log-pow
2×exp-diff
1×rem-exp-log
1×rem-log-exp
1×diff-log
1×pow1/3
1×sub-neg
Counts
4 → 57
Calls
4 calls:
Slowest
602.0ms
(log (- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x)))
521.0ms
(exp (log (- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x))))
236.0ms
(- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x))
0.0ms
(cbrt x)

series627.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
275.0ms
(cbrt x)
119.0ms
(log (- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x)))
117.0ms
(exp (log (- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x))))
116.0ms
(- (cbrt (* (* (* (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1)))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (* (cbrt (+ x 1)) (cbrt (+ x 1))))) (cbrt (+ x 1)))) (cbrt x))

simplify2.7s

Counts
36 → 69
Calls
36 calls:
Slowest
292.0ms
(- (+ (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)))))
271.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
265.0ms
(- (+ (* 1/3 x) 1) (+ (* 1/9 (pow x 2)) (pow x 1/3)))
260.0ms
(- (+ (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)))))
153.0ms
(- (+ (log (* 1/3 (pow x 1/3))) (+ (log (/ 1 x)) (* 7/54 (/ 1 (pow x 2))))) (* 1/3 (/ 1 x)))

prune1.1s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 1.9b

regimes94.0ms

Accuracy

99.8% (0.1b remaining)

Error of 2.2b against oracle of 2.1b and baseline of 29.1b

bsearch207.0ms

end0.0ms

sample2.4s

Algorithm
intervals
Results
737.0ms4181×body80valid
503.0ms1648×body1280valid
462.0ms8053×body80nan
306.0ms1222×body640valid
151.0ms651×body320valid
32.0ms299×body160valid