Average Error: 14.8 → 0.1
Time: 17.9s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\left(\left(\sqrt[3]{1} \cdot \sqrt{1}\right) \cdot \left(-\sqrt[3]{1}\right)\right) \cdot \frac{\frac{\sqrt{1}}{1 + x}}{\frac{x}{\sqrt[3]{1}}}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\left(\left(\sqrt[3]{1} \cdot \sqrt{1}\right) \cdot \left(-\sqrt[3]{1}\right)\right) \cdot \frac{\frac{\sqrt{1}}{1 + x}}{\frac{x}{\sqrt[3]{1}}}
double f(double x) {
        double r50450 = 1.0;
        double r50451 = x;
        double r50452 = r50451 + r50450;
        double r50453 = r50450 / r50452;
        double r50454 = r50450 / r50451;
        double r50455 = r50453 - r50454;
        return r50455;
}
↓
double f(double x) {
        double r50456 = 1.0;
        double r50457 = cbrt(r50456);
        double r50458 = sqrt(r50456);
        double r50459 = r50457 * r50458;
        double r50460 = -r50457;
        double r50461 = r50459 * r50460;
        double r50462 = x;
        double r50463 = r50456 + r50462;
        double r50464 = r50458 / r50463;
        double r50465 = r50462 / r50457;
        double r50466 = r50464 / r50465;
        double r50467 = r50461 * r50466;
        return r50467;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.8

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

  3. Applied frac-sub14.2

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

  4. Simplified14.2

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

  5. Simplified14.2

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

  7. Applied associate-/r*14.2

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

  8. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{0 - 1}{\frac{x}{1}}}}{x + 1}\]
  9. Using strategy rm

  10. Applied *-un-lft-identity0.1

    \[\leadsto \frac{\frac{0 - 1}{\frac{x}{1}}}{\color{blue}{1 \cdot \left(x + 1\right)}}\]

  11. Applied add-cube-cbrt0.1

    \[\leadsto \frac{\frac{0 - 1}{\frac{x}{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}}}{1 \cdot \left(x + 1\right)}\]

  12. Applied *-un-lft-identity0.1

    \[\leadsto \frac{\frac{0 - 1}{\frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}}{1 \cdot \left(x + 1\right)}\]

  13. Applied times-frac0.1

    \[\leadsto \frac{\frac{0 - 1}{\color{blue}{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \frac{x}{\sqrt[3]{1}}}}}{1 \cdot \left(x + 1\right)}\]

  14. Applied add-sqr-sqrt0.1

    \[\leadsto \frac{\frac{0 - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \frac{x}{\sqrt[3]{1}}}}{1 \cdot \left(x + 1\right)}\]

  15. Applied add-sqr-sqrt0.1

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{0} \cdot \sqrt{0}} - \sqrt{1} \cdot \sqrt{1}}{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \frac{x}{\sqrt[3]{1}}}}{1 \cdot \left(x + 1\right)}\]

  16. Applied difference-of-squares0.1

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt{0} + \sqrt{1}\right) \cdot \left(\sqrt{0} - \sqrt{1}\right)}}{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \frac{x}{\sqrt[3]{1}}}}{1 \cdot \left(x + 1\right)}\]

  17. Applied times-frac0.1

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{0} + \sqrt{1}}{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}}} \cdot \frac{\sqrt{0} - \sqrt{1}}{\frac{x}{\sqrt[3]{1}}}}}{1 \cdot \left(x + 1\right)}\]

  18. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{0} + \sqrt{1}}{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}}{1} \cdot \frac{\frac{\sqrt{0} - \sqrt{1}}{\frac{x}{\sqrt[3]{1}}}}{x + 1}}\]

  19. Simplified0.1

    \[\leadsto \color{blue}{\left(\left(\sqrt{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}\right)} \cdot \frac{\frac{\sqrt{0} - \sqrt{1}}{\frac{x}{\sqrt[3]{1}}}}{x + 1}\]

  20. Simplified0.1

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

  21. Final simplification0.1

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))