Average Error: 20.1 → 0.6
Time: 16.1s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x + 1} \cdot \sqrt{x}\right) + \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x} \cdot x + \left(x + 1\right) \cdot \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x + 1} \cdot \sqrt{x}\right) + \sqrt{x} \cdot \sqrt{x}\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 r5033031 = 1.0;
        double r5033032 = x;
        double r5033033 = sqrt(r5033032);
        double r5033034 = r5033031 / r5033033;
        double r5033035 = r5033032 + r5033031;
        double r5033036 = sqrt(r5033035);
        double r5033037 = r5033031 / r5033036;
        double r5033038 = r5033034 - r5033037;
        return r5033038;
}
↓
double f(double x) {
        double r5033039 = x;
        double r5033040 = 1.0;
        double r5033041 = r5033039 + r5033040;
        double r5033042 = sqrt(r5033041);
        double r5033043 = r5033042 * r5033042;
        double r5033044 = sqrt(r5033039);
        double r5033045 = r5033042 * r5033044;
        double r5033046 = r5033043 - r5033045;
        double r5033047 = r5033044 * r5033044;
        double r5033048 = r5033046 + r5033047;
        double r5033049 = r5033044 * r5033039;
        double r5033050 = r5033041 * r5033042;
        double r5033051 = r5033049 + r5033050;
        double r5033052 = r5033040 / r5033051;
        double r5033053 = r5033048 * r5033052;
        double r5033054 = r5033053 / r5033045;
        return r5033054;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.6
Herbie0.6
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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. Using strategy rm

  5. Applied flip--19.8

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

  6. Simplified19.4

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

  7. Simplified19.4

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

  9. Applied flip3-+19.4

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

  10. Applied associate-/r/19.4

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

  11. Simplified0.6

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

  12. Final simplification0.6

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

Reproduce

herbie shell --seed 2019162 
(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)))))