Average Error: 19.8 → 0.4
Time: 5.9s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{x + 1}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\left(\sqrt{x} \cdot \sqrt{\sqrt{x + 1}}\right) \cdot \sqrt{\sqrt{x + 1}}}
double f(double x) {
        double r194003 = 1.0;
        double r194004 = x;
        double r194005 = sqrt(r194004);
        double r194006 = r194003 / r194005;
        double r194007 = r194004 + r194003;
        double r194008 = sqrt(r194007);
        double r194009 = r194003 / r194008;
        double r194010 = r194006 - r194009;
        return r194010;
}
↓
double f(double x) {
        double r194011 = 1.0;
        double r194012 = x;
        double r194013 = r194012 + r194011;
        double r194014 = sqrt(r194013);
        double r194015 = sqrt(r194012);
        double r194016 = r194014 + r194015;
        double r194017 = r194011 / r194016;
        double r194018 = r194011 * r194017;
        double r194019 = sqrt(r194014);
        double r194020 = r194015 * r194019;
        double r194021 = r194020 * r194019;
        double r194022 = r194018 / r194021;
        return r194022;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Simplified19.8

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

  6. Applied flip--19.6

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

  7. Simplified19.1

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

  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 add-sqr-sqrt0.4

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

  11. Applied sqrt-prod0.4

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

  12. Applied associate-*r*0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019353 
(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)))))