Average Error: 30.1 → 0.5
Time: 13.3s
Precision: binary64
Cost: 32960
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (/
  1.0
  (+
   (* (cbrt x) (cbrt x))
   (* (cbrt (+ 1.0 x)) (+ (cbrt x) (cbrt (+ 1.0 x)))))))
double code(double x) {
	return cbrt(x + 1.0) - cbrt(x);
}
↓
double code(double x) {
	return 1.0 / ((cbrt(x) * cbrt(x)) + (cbrt(1.0 + x) * (cbrt(x) + cbrt(1.0 + x))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.7
Cost97408
\[\frac{1}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left({\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{2}}\right)}^{4} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}^{4}\right) + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\]
Alternative 2
Error0.9
Cost90944
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}^{4} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}^{8}\right)}\]
Alternative 3
Error0.9
Cost90944
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)}^{2}\right)}^{4}}\]
Alternative 4
Error32.2
Cost90944
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left({\left(\sqrt[3]{\sqrt{\sqrt[3]{x}}}\right)}^{4} \cdot {\left(\sqrt[3]{\sqrt{\sqrt[3]{x}}}\right)}^{4}\right)}\]
Alternative 5
Error32.1
Cost90944
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left({\left(\sqrt[3]{\sqrt[3]{\sqrt{x}}}\right)}^{4} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{x}}}\right)}^{4}\right)}\]
Alternative 6
Error32.1
Cost84480
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{x}}}\right)}^{4}}\]
Alternative 7
Error32.1
Cost84480
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot {\left(\sqrt{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{x}}}\right)}^{4}}\]
Alternative 8
Error32.0
Cost71616
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right) \cdot \left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right)}\]
Alternative 9
Error32.0
Cost71616
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \left(\sqrt{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt{x}}\right) \cdot \left(\sqrt{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt{x}}\right)}\]
Alternative 10
Error32.6
Cost71552
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot {\left(e^{\log \left(\sqrt[3]{\sqrt[3]{x}}\right)}\right)}^{4}}\]
Alternative 11
Error0.7
Cost65152
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}\]
Alternative 12
Error1.6
Cost58816
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{0.3333333333333333}}\]
Alternative 13
Error0.6
Cost58752
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4}}\]
Alternative 14
Error0.6
Cost58688
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4} \cdot \sqrt[3]{{\left(\sqrt[3]{x}\right)}^{2}}}\]
Alternative 15
Error30.2
Cost58688
\[\sqrt[3]{\sqrt[3]{1 + x} - \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} - \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{1 + x} - \sqrt[3]{x}}\right)\]
Alternative 16
Error43.0
Cost53504
\[\frac{1}{{x}^{0.6666666666666666} + \frac{\sqrt[3]{1 + {x}^{3}} \cdot \left(x + \left(1 + x\right)\right)}{\sqrt[3]{x \cdot x + \left(1 - x\right)} \cdot \left({x}^{0.6666666666666666} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)\right)}}\]
Alternative 17
Error7.9
Cost52416
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{x \cdot \sqrt[3]{x}}}\]
Alternative 18
Error32.6
Cost52352
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{{x}^{0.6666666666666666}}\right)}\]
Alternative 19
Error15.8
Cost52352
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4} \cdot \sqrt[3]{\sqrt[3]{x \cdot x}}}\]
Alternative 20
Error32.1
Cost52288
\[\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} + \sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}\right)}\]
Alternative 21
Error32.0
Cost52288
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{\sqrt{x}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt{x}}\right)}\]
Alternative 22
Error32.1
Cost52224
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt{\sqrt[3]{x}} \cdot {\left(\sqrt{\sqrt[3]{x}}\right)}^{3}}\]
Alternative 23
Error0.7
Cost52224
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{\sqrt[3]{x}} \cdot {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{5}}\]
Alternative 24
Error40.0
Cost46912
\[\frac{1}{{x}^{0.6666666666666666} + \frac{\sqrt[3]{x \cdot x + -1} \cdot \left(x + \left(1 + x\right)\right)}{\sqrt[3]{x + -1} \cdot \left({x}^{0.6666666666666666} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)\right)}}\]
Alternative 25
Error47.4
Cost46592
\[\left(0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} + 0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\right) - \left(0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + 0.0411522633744856 \cdot \sqrt[3]{\frac{1}{{x}^{11}}}\right)\]
Alternative 26
Error16.1
Cost46016
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + {\left(x \cdot x\right)}^{0.1111111111111111} \cdot {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{4}}\]
Alternative 27
Error32.9
Cost45888
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + {\left(\sqrt{x}\right)}^{0.6666666666666666} \cdot {\left(\sqrt{x}\right)}^{0.6666666666666666}}\]
Alternative 28
Error32.5
Cost45760
\[\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} + e^{\log \left(\sqrt[3]{x}\right)}\right)}\]
Alternative 29
Error32.2
Cost45760
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + {\left({x}^{\left(\sqrt{0.6666666666666666}\right)}\right)}^{\left(\sqrt{0.6666666666666666}\right)}}\]
Alternative 30
Error30.3
Cost45632
\[\sqrt[3]{\sqrt[3]{1 + x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right) - \sqrt[3]{x}\]
Alternative 31
Error30.3
Cost45376
\[\sqrt[3]{1 + x} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\]
Alternative 32
Error32.9
Cost39296
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} + e^{\log \left(\sqrt[3]{x}\right)}\right) + {x}^{0.6666666666666666}}\]
Alternative 33
Error30.2
Cost39232
\[\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}} - \sqrt[3]{x}\]
Alternative 34
Error47.2
Cost33344
\[0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \left(0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right)\]
Alternative 35
Error32.9
Cost32960
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + e^{0.6666666666666666 \cdot \log x}}\]
Alternative 36
Error0.5
Cost32896
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + {\left(\sqrt[3]{x}\right)}^{2}}\]
Alternative 37
Error15.7
Cost26560
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + \sqrt[3]{x \cdot x}}\]
Alternative 38
Error31.0
Cost26496
\[\frac{\sqrt[3]{1 + {x}^{3}}}{\sqrt[3]{x \cdot x + \left(1 - x\right)}} - \sqrt[3]{x}\]
Alternative 39
Error32.9
Cost26496
\[\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) + {x}^{0.6666666666666666}}\]
Alternative 40
Error30.1
Cost25984
\[\sqrt[3]{{\left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)}^{3}}\]
Alternative 41
Error30.1
Cost25920
\[e^{\log \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)}\]
Alternative 42
Error30.1
Cost25920
\[\log \left(e^{\sqrt[3]{1 + x} - \sqrt[3]{x}}\right)\]
Alternative 43
Error47.4
Cost20096
\[0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\]
Alternative 44
Error30.8
Cost19904
\[\frac{\sqrt[3]{x \cdot x + -1}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\]
Alternative 45
Error31.5
Cost13184
\[{\left(1 + x\right)}^{0.3333333333333333} - \sqrt[3]{x}\]
Alternative 46
Error30.1
Cost13120
\[\sqrt[3]{1 + x} - \sqrt[3]{x}\]
Alternative 47
Error31.7
Cost6848
\[\left(1 - \sqrt[3]{x}\right) + x \cdot 0.3333333333333333\]
Alternative 48
Error46.8
Cost6848
\[0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}\]
Alternative 49
Error31.8
Cost6592
\[1 - \sqrt[3]{x}\]
Alternative 50
Error61.7
Cost64
\[0\]
Alternative 51
Error32.3
Cost64
\[1\]
Alternative 52
Error62.9
Cost64
\[-1\]

Error

Derivation

  1. Initial program 30.1

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

  3. Applied flip3--_binary64_7530.0

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

  4. Simplified0.5

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

  5. Simplified32.9

    \[\leadsto \frac{1}{\color{blue}{{x}^{0.6666666666666666} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}}\]
  6. Using strategy rm

  7. Applied sqr-pow_binary64_4332.9

    \[\leadsto \frac{1}{\color{blue}{{x}^{\left(\frac{0.6666666666666666}{2}\right)} \cdot {x}^{\left(\frac{0.6666666666666666}{2}\right)}} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\]

  8. Simplified32.7

    \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x}} \cdot {x}^{\left(\frac{0.6666666666666666}{2}\right)} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\]

  9. Simplified0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2021042 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))