Average Error: 19.6 → 0.3
Time: 14.1s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1 \cdot 1}{x + \sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1 \cdot 1}{x + \sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{x + 1}}
double f(double x) {
        double r111671 = 1.0;
        double r111672 = x;
        double r111673 = sqrt(r111672);
        double r111674 = r111671 / r111673;
        double r111675 = r111672 + r111671;
        double r111676 = sqrt(r111675);
        double r111677 = r111671 / r111676;
        double r111678 = r111674 - r111677;
        return r111678;
}
↓
double f(double x) {
        double r111679 = 1.0;
        double r111680 = r111679 * r111679;
        double r111681 = x;
        double r111682 = r111681 + r111679;
        double r111683 = sqrt(r111682);
        double r111684 = sqrt(r111681);
        double r111685 = r111683 * r111684;
        double r111686 = r111681 + r111685;
        double r111687 = r111680 / r111686;
        double r111688 = r111687 / r111683;
        return r111688;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.6

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

  3. Applied frac-sub19.6

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

  4. Simplified19.6

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

  6. Applied flip--19.5

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

  7. Simplified19.1

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

  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

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 
(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)))))