Average Error: 14.1 → 0.1
Time: 6.8s
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
Accuracy0.3
Cost1280
\[\frac{1}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \frac{\frac{-1}{x}}{\sqrt[3]{1 + x}}\]
Alternative 2
Accuracy1.1
Cost1856
\[\frac{\sqrt[3]{\frac{-1}{x}} \cdot \sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{x + 1}}\]
Alternative 3
Accuracy1.2
Cost2048
\[\frac{\frac{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \frac{\frac{\sqrt[3]{-1}}{\sqrt[3]{x}}}{\sqrt[3]{x + 1}}\]

Derivation

  1. Initial program 14.1

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

  3. Applied frac-sub_binary64_76913.4

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

  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_7040.1

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

  9. Applied pow1_binary64_8210.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020338 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))