Average Error: 19.4 → 0.3
Time: 14.8s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\frac{\sqrt{x}}{1}}}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\frac{\sqrt{x}}{1}}}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r120619 = 1.0;
        double r120620 = x;
        double r120621 = sqrt(r120620);
        double r120622 = r120619 / r120621;
        double r120623 = r120620 + r120619;
        double r120624 = sqrt(r120623);
        double r120625 = r120619 / r120624;
        double r120626 = r120622 - r120625;
        return r120626;
}
↓
double f(double x) {
        double r120627 = 1.0;
        double r120628 = x;
        double r120629 = sqrt(r120628);
        double r120630 = r120629 / r120627;
        double r120631 = r120627 / r120630;
        double r120632 = r120628 + r120627;
        double r120633 = sqrt(r120632);
        double r120634 = r120629 * r120633;
        double r120635 = r120632 + r120634;
        double r120636 = r120631 / r120635;
        return r120636;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.4
Target0.6
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.4

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

  3. Applied frac-sub19.4

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

  4. Simplified19.4

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

  6. Applied flip--19.2

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

  7. Simplified18.8

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

  8. Taylor expanded around 0 0.4

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

  10. Applied add-sqr-sqrt0.5

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1)))))

  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))