Average Error: 19.6 → 0.7
Time: 9.4s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot 1}{\sqrt{x + 1} \cdot x + \left(x + 1\right) \cdot \sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot 1}{\sqrt{x + 1} \cdot x + \left(x + 1\right) \cdot \sqrt{x}}
double f(double x) {
        double r126180 = 1.0;
        double r126181 = x;
        double r126182 = sqrt(r126181);
        double r126183 = r126180 / r126182;
        double r126184 = r126181 + r126180;
        double r126185 = sqrt(r126184);
        double r126186 = r126180 / r126185;
        double r126187 = r126183 - r126186;
        return r126187;
}
↓
double f(double x) {
        double r126188 = 1.0;
        double r126189 = r126188 * r126188;
        double r126190 = x;
        double r126191 = r126190 + r126188;
        double r126192 = sqrt(r126191);
        double r126193 = r126192 * r126190;
        double r126194 = sqrt(r126190);
        double r126195 = r126191 * r126194;
        double r126196 = r126193 + r126195;
        double r126197 = r126189 / r126196;
        return r126197;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.6
Target0.7
Herbie0.7
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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

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

  6. Applied flip--19.4

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

  7. Applied associate-*r/19.4

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

  8. Applied associate-/l/19.4

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

  9. Simplified19.4

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

  10. Taylor expanded around 0 0.7

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

  12. Applied *-un-lft-identity0.7

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

  13. Final simplification0.7

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

Reproduce

herbie shell --seed 2020046 +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)))))