Average Error: 19.8 → 10.1
Time: 20.8s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 9323.6116085612:\\ \;\;\;\;{x}^{\frac{-1}{2}} - \frac{1}{\sqrt{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{{x}^{7}}} \cdot \frac{5}{16} - \sqrt{\frac{1}{{x}^{5}}} \cdot \frac{3}{8}\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\\ \end{array}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 9323.6116085612:\\
\;\;\;\;{x}^{\frac{-1}{2}} - \frac{1}{\sqrt{1 + x}}\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt{\frac{1}{{x}^{7}}} \cdot \frac{5}{16} - \sqrt{\frac{1}{{x}^{5}}} \cdot \frac{3}{8}\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\\

\end{array}
double f(double x) {
        double r4300113 = 1.0;
        double r4300114 = x;
        double r4300115 = sqrt(r4300114);
        double r4300116 = r4300113 / r4300115;
        double r4300117 = r4300114 + r4300113;
        double r4300118 = sqrt(r4300117);
        double r4300119 = r4300113 / r4300118;
        double r4300120 = r4300116 - r4300119;
        return r4300120;
}
↓
double f(double x) {
        double r4300121 = x;
        double r4300122 = 9323.6116085612;
        bool r4300123 = r4300121 <= r4300122;
        double r4300124 = -0.5;
        double r4300125 = pow(r4300121, r4300124);
        double r4300126 = 1.0;
        double r4300127 = r4300126 + r4300121;
        double r4300128 = sqrt(r4300127);
        double r4300129 = r4300126 / r4300128;
        double r4300130 = r4300125 - r4300129;
        double r4300131 = 7.0;
        double r4300132 = pow(r4300121, r4300131);
        double r4300133 = r4300126 / r4300132;
        double r4300134 = sqrt(r4300133);
        double r4300135 = 0.3125;
        double r4300136 = r4300134 * r4300135;
        double r4300137 = 5.0;
        double r4300138 = pow(r4300121, r4300137);
        double r4300139 = r4300126 / r4300138;
        double r4300140 = sqrt(r4300139);
        double r4300141 = 0.375;
        double r4300142 = r4300140 * r4300141;
        double r4300143 = r4300136 - r4300142;
        double r4300144 = 0.5;
        double r4300145 = r4300121 * r4300121;
        double r4300146 = r4300121 * r4300145;
        double r4300147 = r4300126 / r4300146;
        double r4300148 = sqrt(r4300147);
        double r4300149 = r4300144 * r4300148;
        double r4300150 = r4300143 + r4300149;
        double r4300151 = r4300123 ? r4300130 : r4300150;
        return r4300151;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.7
Herbie10.1
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 9323.6116085612

    1. Initial program 0.3

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

    3. Applied pow10.3

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

    4. Applied sqrt-pow10.3

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

    5. Applied pow-flip0.1

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

    6. Simplified0.1

      \[\leadsto {x}^{\color{blue}{\frac{-1}{2}}} - \frac{1}{\sqrt{x + 1}}\]

    if 9323.6116085612 < x

    1. Initial program 39.2

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

    3. Applied add-sqr-sqrt49.0

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

    4. Applied add-sqr-sqrt39.2

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

    5. Applied difference-of-squares39.2

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

    6. Taylor expanded around inf 20.1

      \[\leadsto \color{blue}{\left(\frac{5}{16} \cdot \sqrt{\frac{1}{{x}^{7}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) - \frac{3}{8} \cdot \sqrt{\frac{1}{{x}^{5}}}}\]

    7. Simplified20.1

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\left(x \cdot x\right) \cdot x}} \cdot \frac{1}{2} + \left(\sqrt{\frac{1}{{x}^{7}}} \cdot \frac{5}{16} - \frac{3}{8} \cdot \sqrt{\frac{1}{{x}^{5}}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 9323.6116085612:\\ \;\;\;\;{x}^{\frac{-1}{2}} - \frac{1}{\sqrt{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{{x}^{7}}} \cdot \frac{5}{16} - \sqrt{\frac{1}{{x}^{5}}} \cdot \frac{3}{8}\right) + \frac{1}{2} \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(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)))))