Average Error: 19.7 → 0.7
Time: 17.4s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot \left(x \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot \left(x \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right)\right)}
double f(double x) {
        double r1876154 = 1.0;
        double r1876155 = x;
        double r1876156 = sqrt(r1876155);
        double r1876157 = r1876154 / r1876156;
        double r1876158 = r1876155 + r1876154;
        double r1876159 = sqrt(r1876158);
        double r1876160 = r1876154 / r1876159;
        double r1876161 = r1876157 - r1876160;
        return r1876161;
}
↓
double f(double x) {
        double r1876162 = 1.0;
        double r1876163 = x;
        double r1876164 = r1876163 + r1876162;
        double r1876165 = sqrt(r1876164);
        double r1876166 = r1876165 * r1876165;
        double r1876167 = sqrt(r1876163);
        double r1876168 = r1876162 / r1876167;
        double r1876169 = r1876162 / r1876165;
        double r1876170 = r1876168 + r1876169;
        double r1876171 = r1876163 * r1876170;
        double r1876172 = r1876166 * r1876171;
        double r1876173 = r1876162 / r1876172;
        return r1876173;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 19.7

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

  3. Applied flip--19.8

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

  5. Applied frac-times24.7

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

  6. Applied frac-times19.9

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

  7. Applied frac-sub19.6

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

  8. Applied associate-/l/19.6

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

  9. Taylor expanded around 0 5.4

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

  11. Applied associate-*r*0.9

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

  12. Simplified0.7

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

  13. Final simplification0.7

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

Reproduce

herbie shell --seed 2019128 
(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)))))