Average Error: 20.0 → 0.3
Time: 9.2s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1 \cdot 1}{x + \sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1 \cdot 1}{x + \sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1}}
double f(double x) {
        double r145585 = 1.0;
        double r145586 = x;
        double r145587 = sqrt(r145586);
        double r145588 = r145585 / r145587;
        double r145589 = r145586 + r145585;
        double r145590 = sqrt(r145589);
        double r145591 = r145585 / r145590;
        double r145592 = r145588 - r145591;
        return r145592;
}
↓
double f(double x) {
        double r145593 = 1.0;
        double r145594 = r145593 * r145593;
        double r145595 = x;
        double r145596 = sqrt(r145595);
        double r145597 = r145595 + r145593;
        double r145598 = sqrt(r145597);
        double r145599 = r145596 * r145598;
        double r145600 = r145595 + r145599;
        double r145601 = r145594 / r145600;
        double r145602 = r145601 / r145598;
        return r145602;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.0

    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
  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. Simplified20.0

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

  6. Applied flip--19.7

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

  7. Simplified19.3

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

  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 associate-/r*0.4

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 
(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)))))