Average Error: 19.6 → 0.4
Time: 17.3s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}}{\sqrt{x + 1} \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1}}{\sqrt{x + 1} \cdot \sqrt{x}}
double f(double x) {
        double r107427 = 1.0;
        double r107428 = x;
        double r107429 = sqrt(r107428);
        double r107430 = r107427 / r107429;
        double r107431 = r107428 + r107427;
        double r107432 = sqrt(r107431);
        double r107433 = r107427 / r107432;
        double r107434 = r107430 - r107433;
        return r107434;
}
↓
double f(double x) {
        double r107435 = 1.0;
        double r107436 = x;
        double r107437 = r107436 + r107435;
        double r107438 = sqrt(r107437);
        double r107439 = sqrt(r107436);
        double r107440 = r107438 + r107439;
        double r107441 = r107440 * r107435;
        double r107442 = r107435 / r107441;
        double r107443 = r107438 * r107439;
        double r107444 = r107442 / r107443;
        return r107444;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 19.6

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

  3. Applied frac-sub19.6

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

  4. Simplified19.6

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

  6. Applied flip--19.5

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

  7. Simplified19.1

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

  8. Simplified19.1

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

  9. Taylor expanded around 0 0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(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)))))