Average Error: 19.3 → 5.3
Time: 29.6s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{\left(1 \cdot 1\right) \cdot 1}{x \cdot \left(x + 1\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{\left(1 \cdot 1\right) \cdot 1}{x \cdot \left(x + 1\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}
double f(double x) {
        double r165111 = 1.0;
        double r165112 = x;
        double r165113 = sqrt(r165112);
        double r165114 = r165111 / r165113;
        double r165115 = r165112 + r165111;
        double r165116 = sqrt(r165115);
        double r165117 = r165111 / r165116;
        double r165118 = r165114 - r165117;
        return r165118;
}
↓
double f(double x) {
        double r165119 = 1.0;
        double r165120 = r165119 * r165119;
        double r165121 = r165120 * r165119;
        double r165122 = x;
        double r165123 = r165122 + r165119;
        double r165124 = r165122 * r165123;
        double r165125 = r165121 / r165124;
        double r165126 = sqrt(r165122);
        double r165127 = r165119 / r165126;
        double r165128 = sqrt(r165123);
        double r165129 = r165119 / r165128;
        double r165130 = r165127 + r165129;
        double r165131 = r165125 / r165130;
        return r165131;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.3
Target0.6
Herbie5.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.3

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

  3. Applied flip--19.4

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

  4. Simplified25.3

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

  6. Applied frac-times23.5

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

  7. Applied frac-sub19.1

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

  8. Simplified18.8

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

  9. Simplified18.8

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

  10. Taylor expanded around 0 5.3

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

  11. Final simplification5.3

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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