Average Error: 14.1 → 0.1
Time: 2.1s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x} \]
↓
\[\frac{\frac{-1}{x}}{x + 1} \]
\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

Derivation

  1. Initial program 14.1

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

  3. Applied frac-sub_binary6413.5

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  7. Applied *-un-lft-identity_binary640.4

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

  8. Applied distribute-rgt-out_binary640.4

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

  9. Applied *-un-lft-identity_binary640.4

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

  10. Applied times-frac_binary640.1

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

  11. Simplified0.1

    \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{-1}{x + 1}} \]
  12. Using strategy rm

  13. Applied associate-*r/_binary640.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2021198 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))