Average Error: 19.9 → 0.4
Time: 1.2m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[1 \cdot \frac{\frac{\frac{1}{\sqrt{x}}}{\sqrt{x + 1}}}{1 \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
1 \cdot \frac{\frac{\frac{1}{\sqrt{x}}}{\sqrt{x + 1}}}{1 \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}
double f(double x) {
        double r5953419 = 1.0;
        double r5953420 = x;
        double r5953421 = sqrt(r5953420);
        double r5953422 = r5953419 / r5953421;
        double r5953423 = r5953420 + r5953419;
        double r5953424 = sqrt(r5953423);
        double r5953425 = r5953419 / r5953424;
        double r5953426 = r5953422 - r5953425;
        return r5953426;
}
↓
double f(double x) {
        double r5953427 = 1.0;
        double r5953428 = 1.0;
        double r5953429 = x;
        double r5953430 = sqrt(r5953429);
        double r5953431 = r5953428 / r5953430;
        double r5953432 = r5953429 + r5953427;
        double r5953433 = sqrt(r5953432);
        double r5953434 = r5953431 / r5953433;
        double r5953435 = r5953430 + r5953433;
        double r5953436 = r5953427 * r5953435;
        double r5953437 = r5953434 / r5953436;
        double r5953438 = r5953427 * r5953437;
        return r5953438;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.9

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

  3. Applied frac-sub19.9

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

  5. Applied flip--19.7

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

  6. Applied associate-/l/19.7

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

  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 div-inv0.8

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019200 
(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)))))