Average Error: 19.7 → 0.4
Time: 19.2s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x + 1} \cdot \sqrt{x}}
double f(double x) {
        double r6392427 = 1.0;
        double r6392428 = x;
        double r6392429 = sqrt(r6392428);
        double r6392430 = r6392427 / r6392429;
        double r6392431 = r6392428 + r6392427;
        double r6392432 = sqrt(r6392431);
        double r6392433 = r6392427 / r6392432;
        double r6392434 = r6392430 - r6392433;
        return r6392434;
}
↓
double f(double x) {
        double r6392435 = 1.0;
        double r6392436 = x;
        double r6392437 = sqrt(r6392436);
        double r6392438 = r6392436 + r6392435;
        double r6392439 = sqrt(r6392438);
        double r6392440 = r6392437 + r6392439;
        double r6392441 = r6392435 / r6392440;
        double r6392442 = r6392435 * r6392441;
        double r6392443 = r6392439 * r6392437;
        double r6392444 = r6392442 / r6392443;
        return r6392444;
}

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. Simplified19.7

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

  6. Applied flip--19.5

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

  7. Simplified19.1

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

  8. Taylor expanded around 0 0.4

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

  10. Applied +-commutative0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(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)))))