Average Error: 15.0 → 0.1
Time: 48.7s
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
Error0.4
Cost448
\[\frac{-1}{x + x \cdot x}\]
Alternative 2
Error0.4
Cost448
\[\frac{-1}{x \cdot \left(x + 1\right)}\]
Alternative 3
Error1.1
Cost1090
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0167819694186992:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 1.0086083486822843:\\ \;\;\;\;1 - \left(x + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \end{array}\]
Alternative 4
Error1.2
Cost962
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0167819694186992:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.7684734012841025:\\ \;\;\;\;\frac{-1}{x} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \end{array}\]
Alternative 5
Error1.1
Cost648
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0167819694186992 \lor \neg \left(x \leq 0.7684734012841025\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + 1\\ \end{array}\]
Alternative 6
Error16.5
Cost834
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0167819694186992:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.556559595766306 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 7
Error46.8
Cost64
\[0\]
Alternative 8
Error62.1
Cost64
\[1\]

Error

Time

Derivation

  1. Initial program 15.0

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

  3. Applied frac-sub_binary64_42814.3

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

  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_3630.1

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

  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 2021040 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))