Average Error: 19.3 → 0.3
Time: 15.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1 \cdot 1}{x + \sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1 \cdot 1}{x + \sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{x + 1}}
double f(double x) {
        double r130132 = 1.0;
        double r130133 = x;
        double r130134 = sqrt(r130133);
        double r130135 = r130132 / r130134;
        double r130136 = r130133 + r130132;
        double r130137 = sqrt(r130136);
        double r130138 = r130132 / r130137;
        double r130139 = r130135 - r130138;
        return r130139;
}
↓
double f(double x) {
        double r130140 = 1.0;
        double r130141 = r130140 * r130140;
        double r130142 = x;
        double r130143 = r130142 + r130140;
        double r130144 = sqrt(r130143);
        double r130145 = sqrt(r130142);
        double r130146 = r130144 * r130145;
        double r130147 = r130142 + r130146;
        double r130148 = r130141 / r130147;
        double r130149 = r130148 / r130144;
        return r130149;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 19.3

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

  3. Applied frac-sub19.3

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

  4. Simplified19.3

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

  6. Applied flip--19.1

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

  7. Simplified18.7

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

  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 associate-/r*0.4

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019326 
(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)))))