Average Error: 19.3 → 18.7
Time: 6.1s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \le 1.2087681343018217:\\ \;\;\;\;\frac{{e}^{\left(\log \left(\frac{{1}^{2} \cdot \left(x + 1\right) + \left(-x \cdot {1}^{2}\right)}{1 \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}\right)\right)}}{{e}^{\left(\log \left(\sqrt{x} \cdot \sqrt{x + 1}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x}} - \frac{\frac{1}{\left|\sqrt[3]{x + 1}\right|}}{\sqrt{\sqrt[3]{x + 1}}}\\ \end{array}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \le 1.2087681343018217:\\
\;\;\;\;\frac{{e}^{\left(\log \left(\frac{{1}^{2} \cdot \left(x + 1\right) + \left(-x \cdot {1}^{2}\right)}{1 \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}\right)\right)}}{{e}^{\left(\log \left(\sqrt{x} \cdot \sqrt{x + 1}\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt{x}} - \frac{\frac{1}{\left|\sqrt[3]{x + 1}\right|}}{\sqrt{\sqrt[3]{x + 1}}}\\

\end{array}
double f(double x) {
        double r134108 = 1.0;
        double r134109 = x;
        double r134110 = sqrt(r134109);
        double r134111 = r134108 / r134110;
        double r134112 = r134109 + r134108;
        double r134113 = sqrt(r134112);
        double r134114 = r134108 / r134113;
        double r134115 = r134111 - r134114;
        return r134115;
}
↓
double f(double x) {
        double r134116 = 1.0;
        double r134117 = x;
        double r134118 = sqrt(r134117);
        double r134119 = r134116 / r134118;
        double r134120 = r134117 + r134116;
        double r134121 = sqrt(r134120);
        double r134122 = r134116 / r134121;
        double r134123 = r134119 - r134122;
        double r134124 = 1.2087681343018217;
        bool r134125 = r134123 <= r134124;
        double r134126 = exp(1.0);
        double r134127 = 2.0;
        double r134128 = pow(r134116, r134127);
        double r134129 = r134128 * r134120;
        double r134130 = r134117 * r134128;
        double r134131 = -r134130;
        double r134132 = r134129 + r134131;
        double r134133 = r134118 + r134121;
        double r134134 = r134116 * r134133;
        double r134135 = r134132 / r134134;
        double r134136 = log(r134135);
        double r134137 = pow(r134126, r134136);
        double r134138 = r134118 * r134121;
        double r134139 = log(r134138);
        double r134140 = pow(r134126, r134139);
        double r134141 = r134137 / r134140;
        double r134142 = cbrt(r134120);
        double r134143 = fabs(r134142);
        double r134144 = r134116 / r134143;
        double r134145 = sqrt(r134142);
        double r134146 = r134144 / r134145;
        double r134147 = r134119 - r134146;
        double r134148 = r134125 ? r134141 : r134147;
        return r134148;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.3
Target0.7
Herbie18.7
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))) < 1.2087681343018217

    1. Initial program 38.6

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

    3. Applied add-exp-log38.6

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)}}\]
    4. Using strategy rm

    5. Applied pow138.6

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

    6. Applied log-pow38.6

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

    7. Applied exp-prod38.6

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

    8. Simplified38.6

      \[\leadsto {\color{blue}{e}}^{\left(\log \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)\right)}\]
    9. Using strategy rm

    10. Applied frac-sub38.6

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

    11. Applied log-div38.6

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

    12. Applied pow-sub38.6

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

    14. Applied flip--38.2

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

    15. Simplified37.3

      \[\leadsto \frac{{e}^{\left(\log \left(\frac{\color{blue}{{1}^{2} \cdot \left(x + 1\right) + \left(-x \cdot {1}^{2}\right)}}{1 \cdot \sqrt{x + 1} + \sqrt{x} \cdot 1}\right)\right)}}{{e}^{\left(\log \left(\sqrt{x} \cdot \sqrt{x + 1}\right)\right)}}\]

    16. Simplified37.3

      \[\leadsto \frac{{e}^{\left(\log \left(\frac{{1}^{2} \cdot \left(x + 1\right) + \left(-x \cdot {1}^{2}\right)}{\color{blue}{1 \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}}\right)\right)}}{{e}^{\left(\log \left(\sqrt{x} \cdot \sqrt{x + 1}\right)\right)}}\]

    if 1.2087681343018217 < (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0))))

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.3

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

    4. Applied sqrt-prod0.3

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}}}\]

    5. Applied associate-/r*0.3

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\frac{\frac{1}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}}{\sqrt{\sqrt[3]{x + 1}}}}\]

    6. Simplified0.3

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{\color{blue}{\frac{1}{\left|\sqrt[3]{x + 1}\right|}}}{\sqrt{\sqrt[3]{x + 1}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification18.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \le 1.2087681343018217:\\ \;\;\;\;\frac{{e}^{\left(\log \left(\frac{{1}^{2} \cdot \left(x + 1\right) + \left(-x \cdot {1}^{2}\right)}{1 \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}\right)\right)}}{{e}^{\left(\log \left(\sqrt{x} \cdot \sqrt{x + 1}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x}} - \frac{\frac{1}{\left|\sqrt[3]{x + 1}\right|}}{\sqrt{\sqrt[3]{x + 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(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)))))