Average Error: 19.7 → 0.3
Time: 16.1s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{x + \sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{x + \sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{x + 1}}
double f(double x) {
        double r5393795 = 1.0;
        double r5393796 = x;
        double r5393797 = sqrt(r5393796);
        double r5393798 = r5393795 / r5393797;
        double r5393799 = r5393796 + r5393795;
        double r5393800 = sqrt(r5393799);
        double r5393801 = r5393795 / r5393800;
        double r5393802 = r5393798 - r5393801;
        return r5393802;
}
↓
double f(double x) {
        double r5393803 = 1.0;
        double r5393804 = x;
        double r5393805 = r5393804 + r5393803;
        double r5393806 = sqrt(r5393805);
        double r5393807 = sqrt(r5393804);
        double r5393808 = r5393806 * r5393807;
        double r5393809 = r5393804 + r5393808;
        double r5393810 = r5393803 / r5393809;
        double r5393811 = r5393810 / r5393806;
        return r5393811;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.6
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied frac-sub19.7

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

  5. Applied flip--19.4

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

  6. Simplified18.9

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

  7. Simplified18.9

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

  9. Applied associate-/r*18.9

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 
(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)))))