Average Error: 20.1 → 0.7
Time: 1.5m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x} \cdot \sqrt{x + 1}\right)\right) \cdot \frac{1}{\sqrt{x} \cdot x + \left(x + 1\right) \cdot \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x} \cdot \sqrt{x + 1}\right)\right) \cdot \frac{1}{\sqrt{x} \cdot x + \left(x + 1\right) \cdot \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}
double f(double x) {
        double r4724236 = 1.0;
        double r4724237 = x;
        double r4724238 = sqrt(r4724237);
        double r4724239 = r4724236 / r4724238;
        double r4724240 = r4724237 + r4724236;
        double r4724241 = sqrt(r4724240);
        double r4724242 = r4724236 / r4724241;
        double r4724243 = r4724239 - r4724242;
        return r4724243;
}
↓
double f(double x) {
        double r4724244 = x;
        double r4724245 = 1.0;
        double r4724246 = r4724244 + r4724245;
        double r4724247 = sqrt(r4724246);
        double r4724248 = r4724247 * r4724247;
        double r4724249 = sqrt(r4724244);
        double r4724250 = r4724249 * r4724249;
        double r4724251 = r4724249 * r4724247;
        double r4724252 = r4724250 - r4724251;
        double r4724253 = r4724248 + r4724252;
        double r4724254 = r4724249 * r4724244;
        double r4724255 = r4724246 * r4724247;
        double r4724256 = r4724254 + r4724255;
        double r4724257 = r4724245 / r4724256;
        double r4724258 = r4724253 * r4724257;
        double r4724259 = r4724247 * r4724249;
        double r4724260 = r4724258 / r4724259;
        return r4724260;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.7
Herbie0.7
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.1

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

  3. Applied frac-sub20.0

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

  4. Simplified20.0

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

  6. Applied flip--19.8

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

  7. Simplified0.4

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

  9. Applied flip3-+0.8

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

  10. Applied associate-/r/0.8

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

  11. Simplified0.7

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

  12. Final simplification0.7

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

Reproduce

herbie shell --seed 2019151 
(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)))))