Average Error: 19.8 → 0.5
Time: 7.0s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \left(\frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\right)}{\sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \left(\frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\right)}{\sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r147598 = 1.0;
        double r147599 = x;
        double r147600 = sqrt(r147599);
        double r147601 = r147598 / r147600;
        double r147602 = r147599 + r147598;
        double r147603 = sqrt(r147602);
        double r147604 = r147598 / r147603;
        double r147605 = r147601 - r147604;
        return r147605;
}
↓
double f(double x) {
        double r147606 = 1.0;
        double r147607 = sqrt(r147606);
        double r147608 = x;
        double r147609 = r147608 + r147606;
        double r147610 = sqrt(r147609);
        double r147611 = sqrt(r147608);
        double r147612 = r147610 + r147611;
        double r147613 = sqrt(r147612);
        double r147614 = r147607 / r147613;
        double r147615 = r147614 * r147614;
        double r147616 = r147606 * r147615;
        double r147617 = r147611 * r147610;
        double r147618 = r147616 / r147617;
        return r147618;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.6
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Simplified19.8

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

  6. Applied flip--19.6

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

  7. Simplified19.1

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

  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-sqrt0.4

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

  11. Applied add-sqr-sqrt0.4

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

  12. Applied times-frac0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(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)))))