Average Error: 19.2 → 0.3
Time: 2.7m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\mathsf{fma}\left(\left(\sqrt{x}\right), \left(\sqrt{x + 1}\right), x\right)}}{\sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\mathsf{fma}\left(\left(\sqrt{x}\right), \left(\sqrt{x + 1}\right), x\right)}}{\sqrt{x + 1}}
double f(double x) {
        double r20541839 = 1.0;
        double r20541840 = x;
        double r20541841 = sqrt(r20541840);
        double r20541842 = r20541839 / r20541841;
        double r20541843 = r20541840 + r20541839;
        double r20541844 = sqrt(r20541843);
        double r20541845 = r20541839 / r20541844;
        double r20541846 = r20541842 - r20541845;
        return r20541846;
}
↓
double f(double x) {
        double r20541847 = 1.0;
        double r20541848 = x;
        double r20541849 = sqrt(r20541848);
        double r20541850 = r20541848 + r20541847;
        double r20541851 = sqrt(r20541850);
        double r20541852 = fma(r20541849, r20541851, r20541848);
        double r20541853 = r20541847 / r20541852;
        double r20541854 = r20541853 / r20541851;
        return r20541854;
}

Error

Bits error versus x

Target

Original19.2
Target0.6
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.2

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

  3. Applied frac-sub19.2

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

  4. Simplified19.2

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

  6. Applied flip--19.0

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

  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 associate-/r*0.4

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019125 +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)))))