Average Error: 14.1 → 0.1
Time: 6.2s
Precision: binary64
Cost: 448
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{-1}{x}}{x + 1}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{\frac{-1}{x}}{x + 1}
(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))
↓
(FPCore (x) :precision binary64 (/ (/ -1.0 x) (+ x 1.0)))
double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / x);
}
↓
double code(double x) {
	return (-1.0 / x) / (x + 1.0);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error1.1
Cost39872
\[\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{1 + x}} \cdot \frac{\sqrt[3]{\frac{-1}{x}} \cdot \sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\]
Alternative 2
Error1.1
Cost39680
\[\frac{{\left(\sqrt[3]{\frac{-1}{x}}\right)}^{2}}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{1 + x}}\]
Alternative 3
Error14.7
Cost21056
\[\sqrt[3]{\frac{1}{1 + x} - \frac{1}{x}} \cdot \left(\sqrt[3]{\frac{1}{1 + x} - \frac{1}{x}} \cdot \sqrt[3]{\frac{1}{1 + x} - \frac{1}{x}}\right)\]
Alternative 4
Error0.1
Cost19904
\[\frac{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}{x} \cdot \frac{\sqrt[3]{-1}}{1 + x}\]
Alternative 5
Error32.3
Cost13760
\[{\left(\frac{1}{x}\right)}^{3} - \left(\frac{\frac{1}{x}}{x} + \frac{1}{{x}^{4}}\right)\]
Alternative 6
Error32.6
Cost13504
\[\sqrt{\frac{-1}{x}} \cdot \frac{\sqrt{\frac{-1}{x}}}{1 + x}\]
Alternative 7
Error32.0
Cost13376
\[\frac{-1}{\sqrt{x} \cdot \left(\left(1 + x\right) \cdot \sqrt{x}\right)}\]
Alternative 8
Error31.4
Cost13312
\[\sqrt[3]{{\left(\frac{-1}{x \cdot \left(1 + x\right)}\right)}^{3}}\]
Alternative 9
Error23.4
Cost7296
\[\frac{-1}{x + {x}^{4}} \cdot \left(1 + \left(x \cdot x - x\right)\right)\]
Alternative 10
Error5.1
Cost7040
\[\frac{-1}{x - {x}^{3}} \cdot \left(1 - x\right)\]
Alternative 11
Error32.9
Cost7040
\[{\left(\frac{1}{x}\right)}^{3} + \frac{-1}{x \cdot x}\]
Alternative 12
Error0.1
Cost576
\[\frac{1}{x} \cdot \frac{-1}{1 + x}\]
Alternative 13
Error14.1
Cost576
\[\frac{1}{1 + x} - \frac{1}{x}\]
Alternative 14
Error0.1
Cost448
\[\frac{\frac{-1}{x}}{1 + x}\]
Alternative 15
Error0.4
Cost448
\[\frac{-1}{x \cdot \left(1 + x\right)}\]
Alternative 16
Error32.7
Cost320
\[\frac{-1}{x \cdot x}\]
Alternative 17
Error31.3
Cost320
\[1 + \frac{-1}{x}\]
Alternative 18
Error30.6
Cost192
\[\frac{-1}{x}\]
Alternative 19
Error62.1
Cost64
\[1\]
Alternative 20
Error46.5
Cost64
\[0\]
Alternative 21
Error61.6
Cost64
\[-1\]

Error

Derivation

  1. Initial program 14.1

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

  3. Applied frac-sub_binary64_74613.5

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  7. Applied associate-/r*_binary64_6810.1

    \[\leadsto \color{blue}{\frac{\frac{-1}{x}}{1 + x}}\]

  8. Simplified0.1

    \[\leadsto \color{blue}{\frac{\frac{-1}{x}}{1 + x}}\]

  9. Final simplification0.1

    \[\leadsto \frac{\frac{-1}{x}}{x + 1}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))