Average Error: 19.9 → 9.9
Time: 31.7s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 4508.697553152039:\\ \;\;\;\;{x}^{\frac{-1}{2}} - \frac{\sqrt{\frac{1}{\sqrt{\sqrt{1 + x}}}}}{\sqrt[3]{\sqrt{\sqrt{1 + x}}}} \cdot \frac{\sqrt{\frac{1}{\sqrt{\sqrt{1 + x}}}}}{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{8} \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(\frac{5}{16} \cdot \sqrt{\frac{1}{{x}^{7}}} + \sqrt{\frac{\frac{1}{x}}{x \cdot x}} \cdot \frac{1}{2}\right)\\ \end{array}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 4508.697553152039:\\
\;\;\;\;{x}^{\frac{-1}{2}} - \frac{\sqrt{\frac{1}{\sqrt{\sqrt{1 + x}}}}}{\sqrt[3]{\sqrt{\sqrt{1 + x}}}} \cdot \frac{\sqrt{\frac{1}{\sqrt{\sqrt{1 + x}}}}}{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}}\\

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

\end{array}
double f(double x) {
        double r3975457 = 1.0;
        double r3975458 = x;
        double r3975459 = sqrt(r3975458);
        double r3975460 = r3975457 / r3975459;
        double r3975461 = r3975458 + r3975457;
        double r3975462 = sqrt(r3975461);
        double r3975463 = r3975457 / r3975462;
        double r3975464 = r3975460 - r3975463;
        return r3975464;
}
↓
double f(double x) {
        double r3975465 = x;
        double r3975466 = 4508.697553152039;
        bool r3975467 = r3975465 <= r3975466;
        double r3975468 = -0.5;
        double r3975469 = pow(r3975465, r3975468);
        double r3975470 = 1.0;
        double r3975471 = r3975470 + r3975465;
        double r3975472 = sqrt(r3975471);
        double r3975473 = sqrt(r3975472);
        double r3975474 = r3975470 / r3975473;
        double r3975475 = sqrt(r3975474);
        double r3975476 = cbrt(r3975473);
        double r3975477 = r3975475 / r3975476;
        double r3975478 = r3975476 * r3975476;
        double r3975479 = r3975475 / r3975478;
        double r3975480 = r3975477 * r3975479;
        double r3975481 = r3975469 - r3975480;
        double r3975482 = -0.375;
        double r3975483 = 5.0;
        double r3975484 = pow(r3975465, r3975483);
        double r3975485 = r3975470 / r3975484;
        double r3975486 = sqrt(r3975485);
        double r3975487 = r3975482 * r3975486;
        double r3975488 = 0.3125;
        double r3975489 = 7.0;
        double r3975490 = pow(r3975465, r3975489);
        double r3975491 = r3975470 / r3975490;
        double r3975492 = sqrt(r3975491);
        double r3975493 = r3975488 * r3975492;
        double r3975494 = r3975470 / r3975465;
        double r3975495 = r3975465 * r3975465;
        double r3975496 = r3975494 / r3975495;
        double r3975497 = sqrt(r3975496);
        double r3975498 = 0.5;
        double r3975499 = r3975497 * r3975498;
        double r3975500 = r3975493 + r3975499;
        double r3975501 = r3975487 + r3975500;
        double r3975502 = r3975467 ? r3975481 : r3975501;
        return r3975502;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.7
Herbie9.9
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 4508.697553152039

    1. Initial program 0.3

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

    3. Applied pow1/20.3

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

    4. Applied pow-flip0.1

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

    5. Simplified0.1

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

    7. Applied add-sqr-sqrt0.1

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

    8. Applied sqrt-prod0.1

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

    9. Applied associate-/r*0.1

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

    11. Applied add-cube-cbrt0.1

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

    12. Applied add-sqr-sqrt0.1

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

    13. Applied times-frac0.1

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

    if 4508.697553152039 < x

    1. Initial program 39.6

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

    3. Applied pow1/239.6

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

    4. Applied pow-flip44.6

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

    5. Simplified44.6

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

    7. Applied add-sqr-sqrt50.5

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

    8. Applied add-sqr-sqrt42.0

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

    9. Applied difference-of-squares42.0

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

    10. Taylor expanded around inf 20.6

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

    11. Simplified19.9

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 4508.697553152039:\\ \;\;\;\;{x}^{\frac{-1}{2}} - \frac{\sqrt{\frac{1}{\sqrt{\sqrt{1 + x}}}}}{\sqrt[3]{\sqrt{\sqrt{1 + x}}}} \cdot \frac{\sqrt{\frac{1}{\sqrt{\sqrt{1 + x}}}}}{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{8} \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(\frac{5}{16} \cdot \sqrt{\frac{1}{{x}^{7}}} + \sqrt{\frac{\frac{1}{x}}{x \cdot x}} \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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)))))