Average Error: 19.6 → 0.2
Time: 33.8s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[{x}^{\frac{-1}{2}} \cdot \frac{1}{\mathsf{fma}\left(\sqrt{x}, \sqrt{1 + x}, 1 + x\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
{x}^{\frac{-1}{2}} \cdot \frac{1}{\mathsf{fma}\left(\sqrt{x}, \sqrt{1 + x}, 1 + x\right)}
double f(double x) {
        double r3631163 = 1.0;
        double r3631164 = x;
        double r3631165 = sqrt(r3631164);
        double r3631166 = r3631163 / r3631165;
        double r3631167 = r3631164 + r3631163;
        double r3631168 = sqrt(r3631167);
        double r3631169 = r3631163 / r3631168;
        double r3631170 = r3631166 - r3631169;
        return r3631170;
}
↓
double f(double x) {
        double r3631171 = x;
        double r3631172 = -0.5;
        double r3631173 = pow(r3631171, r3631172);
        double r3631174 = 1.0;
        double r3631175 = sqrt(r3631171);
        double r3631176 = r3631174 + r3631171;
        double r3631177 = sqrt(r3631176);
        double r3631178 = fma(r3631175, r3631177, r3631176);
        double r3631179 = r3631174 / r3631178;
        double r3631180 = r3631173 * r3631179;
        return r3631180;
}

Error

Bits error versus x

Target

Original19.6
Target0.6
Herbie0.2
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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

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

  6. Applied flip--19.4

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

  7. Simplified0.4

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

  9. Applied *-un-lft-identity0.4

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

  10. Applied times-frac0.4

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

  11. Simplified0.3

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

  13. Applied pow1/20.3

    \[\leadsto \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} \cdot \frac{1}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x + 1}, x + 1\right)}\]

  14. Applied pow-flip0.2

    \[\leadsto \color{blue}{{x}^{\left(-\frac{1}{2}\right)}} \cdot \frac{1}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x + 1}, x + 1\right)}\]

  15. Simplified0.2

    \[\leadsto {x}^{\color{blue}{\frac{-1}{2}}} \cdot \frac{1}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x + 1}, x + 1\right)}\]

  16. Final simplification0.2

    \[\leadsto {x}^{\frac{-1}{2}} \cdot \frac{1}{\mathsf{fma}\left(\sqrt{x}, \sqrt{1 + x}, 1 + x\right)}\]

Reproduce

herbie shell --seed 2019139 +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)))))