Average Error: 19.7 → 0.3
Time: 18.0s
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 r4915252 = 1.0;
        double r4915253 = x;
        double r4915254 = sqrt(r4915253);
        double r4915255 = r4915252 / r4915254;
        double r4915256 = r4915253 + r4915252;
        double r4915257 = sqrt(r4915256);
        double r4915258 = r4915252 / r4915257;
        double r4915259 = r4915255 - r4915258;
        return r4915259;
}
↓
double f(double x) {
        double r4915260 = 1.0;
        double r4915261 = x;
        double r4915262 = r4915261 + r4915260;
        double r4915263 = sqrt(r4915262);
        double r4915264 = sqrt(r4915261);
        double r4915265 = r4915263 * r4915264;
        double r4915266 = r4915261 + r4915265;
        double r4915267 = r4915260 / r4915266;
        double r4915268 = r4915267 / r4915263;
        return r4915268;
}

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)))))