Average Error: 19.8 → 0.4
Time: 33.3s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\sqrt{x}}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\frac{1}{\sqrt{x + 1}}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\sqrt{x}}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\frac{1}{\sqrt{x + 1}}}}
double f(double x) {
        double r9303761 = 1.0;
        double r9303762 = x;
        double r9303763 = sqrt(r9303762);
        double r9303764 = r9303761 / r9303763;
        double r9303765 = r9303762 + r9303761;
        double r9303766 = sqrt(r9303765);
        double r9303767 = r9303761 / r9303766;
        double r9303768 = r9303764 - r9303767;
        return r9303768;
}
↓
double f(double x) {
        double r9303769 = 1.0;
        double r9303770 = x;
        double r9303771 = sqrt(r9303770);
        double r9303772 = r9303769 / r9303771;
        double r9303773 = r9303770 + r9303769;
        double r9303774 = sqrt(r9303773);
        double r9303775 = r9303774 + r9303771;
        double r9303776 = r9303769 / r9303774;
        double r9303777 = r9303775 / r9303776;
        double r9303778 = r9303772 / r9303777;
        return r9303778;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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

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

  6. Applied flip--19.6

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

  7. Applied associate-/l/19.6

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

  8. Simplified0.8

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

  10. Applied associate-/r*0.4

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

  12. Applied *-un-lft-identity0.4

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

  13. Applied times-frac0.4

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

  14. Applied associate-/l*0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019121 
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1)))))

  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))