Average Error: 19.9 → 19.7
Time: 21.1s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 5.2088959942045496 \cdot 10^{+107}:\\ \;\;\;\;{x}^{\frac{-1}{2}} - \frac{1}{\sqrt{\sqrt[3]{1 + x}} \cdot \left|\sqrt[3]{1 + x}\right|}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}\right)} \cdot \sqrt[3]{\log \left(\frac{\frac{1}{\sqrt{x} \cdot x} - \frac{1}{\sqrt{1 + x} \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} + \left(\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{1 + x}}}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}\right)}\right)}\\ \end{array}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 5.2088959942045496 \cdot 10^{+107}:\\
\;\;\;\;{x}^{\frac{-1}{2}} - \frac{1}{\sqrt{\sqrt[3]{1 + x}} \cdot \left|\sqrt[3]{1 + x}\right|}\\

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

\end{array}
double f(double x) {
        double r6178367 = 1.0;
        double r6178368 = x;
        double r6178369 = sqrt(r6178368);
        double r6178370 = r6178367 / r6178369;
        double r6178371 = r6178368 + r6178367;
        double r6178372 = sqrt(r6178371);
        double r6178373 = r6178367 / r6178372;
        double r6178374 = r6178370 - r6178373;
        return r6178374;
}
↓
double f(double x) {
        double r6178375 = x;
        double r6178376 = 5.2088959942045496e+107;
        bool r6178377 = r6178375 <= r6178376;
        double r6178378 = -0.5;
        double r6178379 = pow(r6178375, r6178378);
        double r6178380 = 1.0;
        double r6178381 = r6178380 + r6178375;
        double r6178382 = cbrt(r6178381);
        double r6178383 = sqrt(r6178382);
        double r6178384 = fabs(r6178382);
        double r6178385 = r6178383 * r6178384;
        double r6178386 = r6178380 / r6178385;
        double r6178387 = r6178379 - r6178386;
        double r6178388 = sqrt(r6178375);
        double r6178389 = r6178380 / r6178388;
        double r6178390 = sqrt(r6178381);
        double r6178391 = r6178380 / r6178390;
        double r6178392 = r6178389 - r6178391;
        double r6178393 = log(r6178392);
        double r6178394 = cbrt(r6178393);
        double r6178395 = r6178388 * r6178375;
        double r6178396 = r6178380 / r6178395;
        double r6178397 = r6178390 * r6178381;
        double r6178398 = r6178380 / r6178397;
        double r6178399 = r6178396 - r6178398;
        double r6178400 = r6178389 * r6178389;
        double r6178401 = r6178391 + r6178389;
        double r6178402 = r6178401 * r6178391;
        double r6178403 = r6178400 + r6178402;
        double r6178404 = r6178399 / r6178403;
        double r6178405 = log(r6178404);
        double r6178406 = cbrt(r6178405);
        double r6178407 = r6178394 * r6178406;
        double r6178408 = exp(r6178407);
        double r6178409 = pow(r6178408, r6178394);
        double r6178410 = r6178377 ? r6178387 : r6178409;
        return r6178410;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.6
Herbie19.7
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 5.2088959942045496e+107

    1. Initial program 15.1

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

    3. Applied pow1/215.1

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

    4. Applied pow-flip14.9

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

    5. Simplified14.9

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

    7. Applied add-cube-cbrt14.8

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

    8. Applied sqrt-prod14.8

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

    9. Simplified14.8

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

    if 5.2088959942045496e+107 < x

    1. Initial program 29.9

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

    3. Applied add-exp-log29.9

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

    5. Applied add-cube-cbrt29.9

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

    6. Applied exp-prod29.9

      \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\log \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)} \cdot \sqrt[3]{\log \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)}\right)}}\]
    7. Using strategy rm

    8. Applied flip3--29.9

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

    9. Simplified29.9

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

    10. Simplified29.9

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

  4. Final simplification19.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 5.2088959942045496 \cdot 10^{+107}:\\ \;\;\;\;{x}^{\frac{-1}{2}} - \frac{1}{\sqrt{\sqrt[3]{1 + x}} \cdot \left|\sqrt[3]{1 + x}\right|}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}\right)} \cdot \sqrt[3]{\log \left(\frac{\frac{1}{\sqrt{x} \cdot x} - \frac{1}{\sqrt{1 + x} \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} + \left(\frac{1}{\sqrt{1 + x}} + \frac{1}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{1 + x}}}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))))