Average Error: 19.7 → 0.4
Time: 15.8s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{\sqrt{x + 1}}}}{\sqrt{\sqrt{x + 1}}} \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1} + x}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{\sqrt{x + 1}}}}{\sqrt{\sqrt{x + 1}}} \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1} + x}
double f(double x) {
        double r125765 = 1.0;
        double r125766 = x;
        double r125767 = sqrt(r125766);
        double r125768 = r125765 / r125767;
        double r125769 = r125766 + r125765;
        double r125770 = sqrt(r125769);
        double r125771 = r125765 / r125770;
        double r125772 = r125768 - r125771;
        return r125772;
}
↓
double f(double x) {
        double r125773 = 1.0;
        double r125774 = x;
        double r125775 = r125774 + r125773;
        double r125776 = sqrt(r125775);
        double r125777 = sqrt(r125776);
        double r125778 = r125773 / r125777;
        double r125779 = r125778 / r125777;
        double r125780 = sqrt(r125774);
        double r125781 = r125780 * r125776;
        double r125782 = r125781 + r125774;
        double r125783 = r125773 / r125782;
        double r125784 = r125779 * r125783;
        return r125784;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.6
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. Simplified19.7

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

  5. Simplified19.7

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

  7. Applied flip--19.5

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

  8. Simplified0.4

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

  10. Applied times-frac0.4

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

  11. Simplified0.4

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

  13. Applied add-sqr-sqrt0.4

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

  14. Applied sqrt-prod0.4

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

  15. Applied associate-/r*0.4

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

  16. Simplified0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))