Average Error: 20.1 → 0.4
Time: 17.6s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{1 \cdot \sqrt{x + 1} + \sqrt{x} \cdot 1}}{\left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{x + 1}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{1 \cdot \sqrt{x + 1} + \sqrt{x} \cdot 1}}{\left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{x + 1}}}
double f(double x) {
        double r14291796 = 1.0;
        double r14291797 = x;
        double r14291798 = sqrt(r14291797);
        double r14291799 = r14291796 / r14291798;
        double r14291800 = r14291797 + r14291796;
        double r14291801 = sqrt(r14291800);
        double r14291802 = r14291796 / r14291801;
        double r14291803 = r14291799 - r14291802;
        return r14291803;
}
↓
double f(double x) {
        double r14291804 = 1.0;
        double r14291805 = x;
        double r14291806 = r14291805 + r14291804;
        double r14291807 = sqrt(r14291806);
        double r14291808 = r14291804 * r14291807;
        double r14291809 = sqrt(r14291805);
        double r14291810 = r14291809 * r14291804;
        double r14291811 = r14291808 + r14291810;
        double r14291812 = r14291804 / r14291811;
        double r14291813 = sqrt(r14291807);
        double r14291814 = r14291809 * r14291813;
        double r14291815 = r14291814 * r14291813;
        double r14291816 = r14291812 / r14291815;
        return r14291816;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.1

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

  3. Applied frac-sub20.1

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

  5. Applied flip--19.9

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

  6. Taylor expanded around 0 0.4

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

  8. Applied add-sqr-sqrt0.4

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

  9. Applied sqrt-prod0.4

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

  10. Applied associate-*r*0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019173 
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))