Average Error: 19.6 → 0.4
Time: 18.3s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\left(1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\left(1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r129170 = 1.0;
        double r129171 = x;
        double r129172 = sqrt(r129171);
        double r129173 = r129170 / r129172;
        double r129174 = r129171 + r129170;
        double r129175 = sqrt(r129174);
        double r129176 = r129170 / r129175;
        double r129177 = r129173 - r129176;
        return r129177;
}
↓
double f(double x) {
        double r129178 = 1.0;
        double r129179 = x;
        double r129180 = r129179 + r129178;
        double r129181 = sqrt(r129180);
        double r129182 = sqrt(r129179);
        double r129183 = r129181 + r129182;
        double r129184 = r129178 / r129183;
        double r129185 = r129178 * r129184;
        double r129186 = 1.0;
        double r129187 = r129182 * r129181;
        double r129188 = r129186 / r129187;
        double r129189 = r129185 * r129188;
        return r129189;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied frac-sub19.6

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

  4. Simplified19.6

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

  6. Applied flip--19.5

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

  7. Simplified19.1

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

  8. Taylor expanded around 0 0.4

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

  10. Applied div-inv0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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