Average Error: 19.7 → 0.6
Time: 32.2s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\sqrt{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}} \cdot \frac{\sqrt{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\sqrt{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}} \cdot \frac{\sqrt{\frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}}}{\sqrt{\sqrt{x} + \sqrt{x + 1}}}
double f(double x) {
        double r11144989 = 1.0;
        double r11144990 = x;
        double r11144991 = sqrt(r11144990);
        double r11144992 = r11144989 / r11144991;
        double r11144993 = r11144990 + r11144989;
        double r11144994 = sqrt(r11144993);
        double r11144995 = r11144989 / r11144994;
        double r11144996 = r11144992 - r11144995;
        return r11144996;
}
↓
double f(double x) {
        double r11144997 = 1.0;
        double r11144998 = x;
        double r11144999 = r11144998 + r11144997;
        double r11145000 = sqrt(r11144999);
        double r11145001 = sqrt(r11144998);
        double r11145002 = r11145000 * r11145001;
        double r11145003 = r11144997 / r11145002;
        double r11145004 = sqrt(r11145003);
        double r11145005 = r11145001 + r11145000;
        double r11145006 = sqrt(r11145005);
        double r11145007 = r11145004 / r11145006;
        double r11145008 = r11145007 * r11145007;
        return r11145008;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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

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

  6. Applied flip--19.5

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

  7. Applied associate-/l/19.5

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

  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-sqrt0.4

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

  13. Applied add-sqr-sqrt0.6

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

  14. Applied times-frac0.6

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

  15. Final simplification0.6

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

Reproduce

herbie shell --seed 2019104 
(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)))))