Average Error: 19.8 → 0.4
Time: 30.9s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \cdot \frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}} \cdot \frac{1}{\sqrt{x + 1} \cdot \sqrt{x}}
double f(double x) {
        double r12571752 = 1.0;
        double r12571753 = x;
        double r12571754 = sqrt(r12571753);
        double r12571755 = r12571752 / r12571754;
        double r12571756 = r12571753 + r12571752;
        double r12571757 = sqrt(r12571756);
        double r12571758 = r12571752 / r12571757;
        double r12571759 = r12571755 - r12571758;
        return r12571759;
}
↓
double f(double x) {
        double r12571760 = 1.0;
        double r12571761 = x;
        double r12571762 = r12571761 + r12571760;
        double r12571763 = sqrt(r12571762);
        double r12571764 = sqrt(r12571761);
        double r12571765 = r12571763 + r12571764;
        double r12571766 = r12571760 / r12571765;
        double r12571767 = r12571763 * r12571764;
        double r12571768 = r12571760 / r12571767;
        double r12571769 = r12571766 * r12571768;
        return r12571769;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.7
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 *-commutative0.8

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

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

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

  13. Applied times-frac0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019107 
(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)))))