Average Error: 19.6 → 19.3
Time: 23.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.189590584318109 \cdot 10^{+141}:\\ \;\;\;\;{x}^{\frac{-1}{2}} - \sqrt[3]{\frac{\frac{1}{1 + x}}{\sqrt{1 + x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{\frac{-1}{2}}\right) \cdot \left(\left(\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{\frac{-1}{2}}\right) \cdot \left(\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{\frac{-1}{2}}\right)\right)}\\ \end{array}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 1.189590584318109 \cdot 10^{+141}:\\
\;\;\;\;{x}^{\frac{-1}{2}} - \sqrt[3]{\frac{\frac{1}{1 + x}}{\sqrt{1 + x}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\left(\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{\frac{-1}{2}}\right) \cdot \left(\left(\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{\frac{-1}{2}}\right) \cdot \left(\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{\frac{-1}{2}}\right)\right)}\\

\end{array}
double f(double x) {
        double r4227283 = 1.0;
        double r4227284 = x;
        double r4227285 = sqrt(r4227284);
        double r4227286 = r4227283 / r4227285;
        double r4227287 = r4227284 + r4227283;
        double r4227288 = sqrt(r4227287);
        double r4227289 = r4227283 / r4227288;
        double r4227290 = r4227286 - r4227289;
        return r4227290;
}
↓
double f(double x) {
        double r4227291 = x;
        double r4227292 = 1.189590584318109e+141;
        bool r4227293 = r4227291 <= r4227292;
        double r4227294 = -0.5;
        double r4227295 = pow(r4227291, r4227294);
        double r4227296 = 1.0;
        double r4227297 = r4227296 + r4227291;
        double r4227298 = r4227296 / r4227297;
        double r4227299 = sqrt(r4227297);
        double r4227300 = r4227298 / r4227299;
        double r4227301 = cbrt(r4227300);
        double r4227302 = r4227295 - r4227301;
        double r4227303 = sqrt(r4227291);
        double r4227304 = r4227296 / r4227303;
        double r4227305 = pow(r4227297, r4227294);
        double r4227306 = r4227304 - r4227305;
        double r4227307 = r4227306 * r4227306;
        double r4227308 = r4227306 * r4227307;
        double r4227309 = cbrt(r4227308);
        double r4227310 = r4227293 ? r4227302 : r4227309;
        return r4227310;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.7
Herbie19.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 1.189590584318109e+141

    1. Initial program 17.6

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

    3. Applied pow1/217.6

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

    4. Applied pow-flip17.4

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

    5. Simplified17.4

      \[\leadsto {x}^{\color{blue}{\frac{-1}{2}}} - \frac{1}{\sqrt{x + 1}}\]
    6. Using strategy rm

    7. Applied add-cbrt-cube17.3

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

    8. Applied add-cbrt-cube17.3

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

    9. Applied cbrt-undiv17.2

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

    10. Simplified17.2

      \[\leadsto {x}^{\frac{-1}{2}} - \sqrt[3]{\color{blue}{\frac{\frac{1}{x + 1}}{\sqrt{x + 1}}}}\]

    if 1.189590584318109e+141 < x

    1. Initial program 24.9

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

    3. Applied pow1/224.9

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

    4. Applied pow-flip35.1

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

    5. Simplified35.1

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

    7. Applied add-cbrt-cube25.0

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

  4. Final simplification19.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.189590584318109 \cdot 10^{+141}:\\ \;\;\;\;{x}^{\frac{-1}{2}} - \sqrt[3]{\frac{\frac{1}{1 + x}}{\sqrt{1 + x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{\frac{-1}{2}}\right) \cdot \left(\left(\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{\frac{-1}{2}}\right) \cdot \left(\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{\frac{-1}{2}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(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)))))