Average Error: 19.3 → 0.4
Time: 14.7s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\left(\sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}} \cdot \sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}}\right) \cdot \frac{1}{\sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\left(\sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}} \cdot \sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}}\right) \cdot \frac{1}{\sqrt{x}}
double f(double x) {
        double r2247556 = 1.0;
        double r2247557 = x;
        double r2247558 = sqrt(r2247557);
        double r2247559 = r2247556 / r2247558;
        double r2247560 = r2247557 + r2247556;
        double r2247561 = sqrt(r2247560);
        double r2247562 = r2247556 / r2247561;
        double r2247563 = r2247559 - r2247562;
        return r2247563;
}
↓
double f(double x) {
        double r2247564 = 1.0;
        double r2247565 = x;
        double r2247566 = r2247565 + r2247564;
        double r2247567 = sqrt(r2247566);
        double r2247568 = sqrt(r2247565);
        double r2247569 = r2247567 + r2247568;
        double r2247570 = r2247564 / r2247569;
        double r2247571 = r2247570 / r2247567;
        double r2247572 = sqrt(r2247571);
        double r2247573 = r2247572 * r2247572;
        double r2247574 = r2247564 / r2247568;
        double r2247575 = r2247573 * r2247574;
        return r2247575;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.3
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.3

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

  3. Applied frac-sub19.3

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

  4. Simplified19.3

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

  6. Applied flip--19.0

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

  7. Simplified18.6

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

  9. Applied div-inv18.6

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

  10. Applied times-frac18.6

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

  11. Simplified0.4

    \[\leadsto \color{blue}{\frac{1 + 0}{\sqrt{x}}} \cdot \frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}\]
  12. Using strategy rm

  13. Applied add-sqr-sqrt0.4

    \[\leadsto \frac{1 + 0}{\sqrt{x}} \cdot \color{blue}{\left(\sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}} \cdot \sqrt{\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x + 1}}}\right)}\]

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019156 +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)))))