Average Error: 19.7 → 0.4
Time: 17.5s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{1 \cdot \left(\sqrt{\sqrt{x} + \sqrt{x + 1}} \cdot \sqrt{\sqrt{x} + \sqrt{x + 1}}\right)}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{1 \cdot \left(\sqrt{\sqrt{x} + \sqrt{x + 1}} \cdot \sqrt{\sqrt{x} + \sqrt{x + 1}}\right)}}{\sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r159736 = 1.0;
        double r159737 = x;
        double r159738 = sqrt(r159737);
        double r159739 = r159736 / r159738;
        double r159740 = r159737 + r159736;
        double r159741 = sqrt(r159740);
        double r159742 = r159736 / r159741;
        double r159743 = r159739 - r159742;
        return r159743;
}
↓
double f(double x) {
        double r159744 = 1.0;
        double r159745 = x;
        double r159746 = sqrt(r159745);
        double r159747 = r159745 + r159744;
        double r159748 = sqrt(r159747);
        double r159749 = r159746 + r159748;
        double r159750 = sqrt(r159749);
        double r159751 = r159750 * r159750;
        double r159752 = r159744 * r159751;
        double r159753 = r159744 / r159752;
        double r159754 = r159746 * r159748;
        double r159755 = r159753 / r159754;
        return r159755;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied frac-sub19.7

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

  5. Applied flip--19.5

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

  6. Simplified19.0

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

  7. Simplified19.0

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

  8. Taylor expanded around 0 0.4

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

  10. Applied add-sqr-sqrt0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019235 +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)))))