Average Error: 20.1 → 0.6
Time: 13.9s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x + 1} \cdot \sqrt{x}\right) + \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x} \cdot x + \left(x + 1\right) \cdot \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x + 1} \cdot \sqrt{x}\right) + \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x} \cdot x + \left(x + 1\right) \cdot \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}
double f(double x) {
        double r5124502 = 1.0;
        double r5124503 = x;
        double r5124504 = sqrt(r5124503);
        double r5124505 = r5124502 / r5124504;
        double r5124506 = r5124503 + r5124502;
        double r5124507 = sqrt(r5124506);
        double r5124508 = r5124502 / r5124507;
        double r5124509 = r5124505 - r5124508;
        return r5124509;
}
↓
double f(double x) {
        double r5124510 = x;
        double r5124511 = 1.0;
        double r5124512 = r5124510 + r5124511;
        double r5124513 = sqrt(r5124512);
        double r5124514 = r5124513 * r5124513;
        double r5124515 = sqrt(r5124510);
        double r5124516 = r5124513 * r5124515;
        double r5124517 = r5124514 - r5124516;
        double r5124518 = r5124515 * r5124515;
        double r5124519 = r5124517 + r5124518;
        double r5124520 = r5124515 * r5124510;
        double r5124521 = r5124512 * r5124513;
        double r5124522 = r5124520 + r5124521;
        double r5124523 = r5124511 / r5124522;
        double r5124524 = r5124519 * r5124523;
        double r5124525 = r5124524 / r5124516;
        return r5124525;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.6
Herbie0.6
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.1

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

  3. Applied frac-sub20.0

    \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
  4. Using strategy rm

  5. Applied flip--19.8

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

  6. Simplified19.4

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

  7. Simplified19.4

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

  9. Applied flip3-+19.4

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

  10. Applied associate-/r/19.4

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

  11. Simplified0.6

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

  12. Final simplification0.6

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

Reproduce

herbie shell --seed 2019162 
(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)))))