Average Error: 19.7 → 0.4
Time: 37.4s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{x + 1}}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot x}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{x + 1}}{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot x}
double f(double x) {
        double r5658791 = 1.0;
        double r5658792 = x;
        double r5658793 = sqrt(r5658792);
        double r5658794 = r5658791 / r5658793;
        double r5658795 = r5658792 + r5658791;
        double r5658796 = sqrt(r5658795);
        double r5658797 = r5658791 / r5658796;
        double r5658798 = r5658794 - r5658797;
        return r5658798;
}
↓
double f(double x) {
        double r5658799 = 1.0;
        double r5658800 = x;
        double r5658801 = r5658800 + r5658799;
        double r5658802 = r5658799 / r5658801;
        double r5658803 = sqrt(r5658800);
        double r5658804 = r5658799 / r5658803;
        double r5658805 = sqrt(r5658801);
        double r5658806 = r5658799 / r5658805;
        double r5658807 = r5658804 + r5658806;
        double r5658808 = r5658807 * r5658800;
        double r5658809 = r5658802 / r5658808;
        return r5658809;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied flip--19.7

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

  5. Applied frac-times24.6

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

  6. Applied frac-times19.8

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

  7. Applied frac-sub19.6

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

  8. Simplified19.2

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

  9. Simplified19.1

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

  10. Taylor expanded around 0 5.4

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

  12. Applied distribute-rgt1-in5.4

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

  13. Applied *-un-lft-identity5.4

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

  14. Applied times-frac5.1

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

  15. Applied associate-/l*0.5

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

  16. Simplified0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1)))))

  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))