Average Error: 20.3 → 0.5
Time: 2.7m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}
double f(double x) {
        double r11252751 = 1.0;
        double r11252752 = x;
        double r11252753 = sqrt(r11252752);
        double r11252754 = r11252751 / r11252753;
        double r11252755 = r11252752 + r11252751;
        double r11252756 = sqrt(r11252755);
        double r11252757 = r11252751 / r11252756;
        double r11252758 = r11252754 - r11252757;
        return r11252758;
}
↓
double f(double x) {
        double r11252759 = 1.0;
        double r11252760 = x;
        double r11252761 = r11252760 + r11252759;
        double r11252762 = sqrt(r11252761);
        double r11252763 = sqrt(r11252760);
        double r11252764 = r11252762 + r11252763;
        double r11252765 = sqrt(r11252764);
        double r11252766 = r11252759 / r11252765;
        double r11252767 = r11252762 * r11252763;
        double r11252768 = r11252759 / r11252767;
        double r11252769 = r11252768 / r11252765;
        double r11252770 = r11252766 * r11252769;
        return r11252770;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.3
Target0.6
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.3

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

  3. Applied frac-sub20.3

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

  4. Simplified20.3

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

  6. Applied flip--20.0

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

  7. Applied associate-/l/20.0

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

  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 add-sqr-sqrt0.5

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

  13. Applied div-inv0.5

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

  14. Applied times-frac0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(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)))))