Average Error: 19.7 → 0.4
Time: 19.9s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \cdot \frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \cdot \frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}
double f(double x) {
        double r6162993 = 1.0;
        double r6162994 = x;
        double r6162995 = sqrt(r6162994);
        double r6162996 = r6162993 / r6162995;
        double r6162997 = r6162994 + r6162993;
        double r6162998 = sqrt(r6162997);
        double r6162999 = r6162993 / r6162998;
        double r6163000 = r6162996 - r6162999;
        return r6163000;
}
↓
double f(double x) {
        double r6163001 = 1.0;
        double r6163002 = x;
        double r6163003 = sqrt(r6163002);
        double r6163004 = 1.0;
        double r6163005 = r6163002 + r6163004;
        double r6163006 = sqrt(r6163005);
        double r6163007 = r6163003 * r6163006;
        double r6163008 = r6163001 / r6163007;
        double r6163009 = r6163006 + r6163003;
        double r6163010 = r6163009 * r6163004;
        double r6163011 = r6163004 / r6163010;
        double r6163012 = r6163008 * r6163011;
        return r6163012;
}

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. Using strategy rm

  5. Applied flip--19.5

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

  6. Applied associate-/l/19.5

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

  7. Taylor expanded around 0 0.8

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

  9. Applied *-un-lft-identity0.8

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

  10. Applied times-frac0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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)))))